## Exercises3.2Curve Sketching

Solve the following problems.

###### 1.

Give an example of a function with one critical point which is also an inflection point. You must provide the equation of your function.

$f(x)=x^3\text{.}$
###### 2.

Give an example of a function that satisfies $f(-1)=f(1)=0$ and $f^\prime(x)>0$ for all $x$ in the domain of $f^\prime\text{.}$

$f(x)=x-\mbox{sign} (x)\text{,}$ $x\not= 0\text{.}$
###### 3.

Determine the value of $a$ so that

\begin{equation*} f(x)=\frac{x^2+ax+5}{x+1} \end{equation*}

has a slant asymptote $y=x+3\text{.}$

$4.$
###### 4.

For what values of the constants $a$ and $b$ does the function $f(x)=\ln a+bx^2-\ln x$ have an extremum value $f(2)=1\text{?}$

$a=2\sqrt{e}\text{,}$ $b=0.125\text{.}$
###### 5.

The function $f(x)=ax^{3}+bx$ has a local extreme value of 2 at $x=1\text{.}$ Determine whether this extremum is a local maximum or local minimum.

Local maximum.
###### 6.

Prove that the function $f(x)=x^{151}+x^{37}+x+3$ has neither a local maximum nor a local minimum.

Observe that $f^\prime(x)>0$ for all $x\in\mathbb{R}\text{.}$

###### 7.

A particle moves along a line with a position function $s(t)$ where $s$ is measured in metres and $t$ in seconds. Four graphs are shown below: one corresponds to the function $s(t)\text{,}$ one to the velocity $v(t)$ of the particle, one to its acceleration $a(t)$ and one is unrelated.

1. Observe that the position function $s$ is already labeled. Identify the graphs of $v(t)$ and $a(t)\text{.}$

2. Find all time intervals when the particle is slowing down and when it is speeding up.

3. Estimate the total distance travelled by the particle over the interval $[1,6]\text{.}$

1. Velocity — bottom left; acceleration — bottom right.

2. Speeding up on $(0,1.5)$ and $(3.2,5)\text{.}$

3. Approximately 5.8 units.

###### 8.

Function $f$ is differentiable everywhere. The graph of $f^\prime$ is depicted in the Figure below. $f^\prime$ is negative and concave down at all points not shown in this graph.

1. Does the function $f$ have a local maximum? If so, determine the approximate coordinate(s) of the local maximum point(s).

2. Does the function $f$ have a local minimum? If so, determine the approximate coordinate(s) of the local minimum point(s).

3. Does the function $f$ have any inflection points? If so, determine the approximate coordinate(s) of the inflection point(s).

4. Determine the interval(s) on which the function $f$ is decreasing.

5. Determine the interval(s) on which $f^{\prime\prime}$ is decreasing.

6. If $f$ is a polynomial function, what is the least possible degree of $f\text{?}$

1. $(3,f(3))\text{.}$

2. $(-1,f(-1))\text{.}$

3. $(-0.5,f(-0.5))\text{,}$ $(0.5,f(0.5))\text{,}$ $(2.3,f(2.3))\text{.}$

4. $(-\infty,-1)$ and $(3,\infty)\text{.}$

5. $(-\infty,0)$ and $(1.5,\infty)\text{.}$

6. $5\text{.}$ Observe that the second derivative has three zeros.

###### 9.

Sketch the graph of $f(x)=3x^4-8x^3+10\text{,}$ after answering the following questions.

1. Where is the graph increasing, and where is decreasing?

2. Where is the graph concave upward, and where is it concave downward?

3. Where are the local minima and local maxima? Establish conclusively that they are local minima and maxima.

4. Where are the inflection points?

5. What happens to $f(x)$ as $x\to \infty$ and as $x\to -\infty\text{.}$

Solution
1. From $f'(x)=12x^2(x-2)$ we conclude that $f'(x)>0$ for $x>2$ and $f'(x)\lt 0$ for $x\lt 2\text{.}$ So $f$ is increasing on $(2,\infty)$ and decreasing on $(-\infty,2)\text{.}$

2. From $f"(x)=12x(3x-4)$ it follows that $f"(x)>0$ for $x\lt 0$ or $\ds x>\frac{4}{3}$ and $f"(x)\lt 0$ for $\ds x\in \left( 0,\frac{4}{3}\right)\text{.}$ Also $f"(x)=0$ for $x=0$ and $\ds x=\frac{4}{3}\text{.}$ Thus $f$ is concave upward on $(-\infty,0)$ and on

3. Critical numbers are $x=0$ and $x=2\text{.}$ Since $f'(x)$ does not change sign at $x=0$ there is no local maximum or minimum there. (Note also that $f"(0)=0$ and that the second derivative test is inconclusive.) Since $f'(x)$ changes from negative to positive at $x=2$ there is a local minimum at $x=2\text{.}$ (Note also that $f"(2)>0\text{,}$ so second derivative test says there is a local minimum.)

4. Inflection points are $(0,10)$ and $\ds \left( \frac{4}{3},f\left( \frac{4}{3}\right) \right)\text{.}$

5. $\ds \lim _{x\to \pm \infty}f(x)= \infty\text{.}$

###### 10.

In this question we consider the function $\ds f(x)=\frac{x-3}{\sqrt{x^2-9}}\text{.}$

1. Find the domain of $f\text{.}$

2. Find the coordinates of all $x$- and $y$-intercepts, if any.

3. Find all horizontal and vertical asymptotes, if any.

4. Find all critical numbers, if any.

5. Find all intervals on which $f$ is increasing and those on which $f$ is decreasing.

6. Find the $(x,y)$ coordinates of all maximum and minimum points, if any.

7. Find all intervals on which $f$ is concave up and those on which $f$ is concave down.

8. Find the $(x,y)$ coordinates of all inflection points, if any.

9. Sketch the graph of $y=f(x)$ using all of the above information. All relevant points must be labeled.

Solution
1. From $x^2-9>0$ it follows that the domain of the function $f$ is the set $(-\infty,-3)\cup(3,\infty)\text{.}$

2. The function is not defined at $x=0\text{,}$ so there is no the $y$-intercept. Note that $f(x)\not= 0$ for all $x$ in the domain of $f\text{.}$

3. From $\ds \lim _{x\to \infty }f(x)=1$ and $\ds \lim _{x\to -\infty }f(x)=-1$ we conclude that there are two horizontal asymptotes, $y=1$ (when $x\to \infty$) and $y=-1$ (when $x\to -\infty$). From $\ds \lim_{x\rightarrow 3^+}f(x)=0$ and $\ds \lim_{x\rightarrow -3^-}f(x)=-\infty$ it follows that there is a vertical asymptote at $x=-3\text{.}$

4. Since, for all $x$ in the domain of $f\text{,}$ $\displaystyle f'(x) = \frac{3(x-3)}{(x^2-9)^{3/2}}\not= 0$ we conclude that there is no critical number for the function $f\text{.}$

5. Note that $f'(x)>0$ for $x>3$ and $f'(x)\lt 0$ for $x\lt -3\text{.}$ Thus $f$ increasing on $(3,\infty )$ and decreasing on $(-\infty ,-3)\text{.}$

6. Since the domain of $f$ is the union of two open intervals and since the function is monotone on each of those intervals, it follows that the function $f$ has neither (local or absolute) a maximum nor a minimum.

7. From $\displaystyle f''(x) =-\frac{6(x-3)(x-\frac{3}{2})}{(x^2-9)^{5/2}}$ it follows that $f''(x)\lt 0$ for all $x$ in the domain of $f\text{.}$ Therefore $f(x)$ is concave downwards on its domain.

8. None.

###### 11.

Given $\ds f(x)=\frac{x^2-1}{x}\text{:}$

1. Find the domain and $x$-intercepts.

2. Find all asymptotes.

3. Determine the intervals on which the function is increasing or decreasing. Find the local maximum and minimum, if they exist.

4. Determine the intervals on which the function is concave upward or downward. Find the inflection points, if they exist.

5. Sketch the graph.

Solution
1. The domain of the function $f$ is the set $\mathbb{R}\backslash \{ 0\}\text{.}$ The $x$-intercepts are $\pm 1\text{.}$ Since 0 not in domain of $f$ there is no $y$-intercept.

2. From $\displaystyle \lim_{x\rightarrow 0^-} f(x) = -\infty$ and $\ds \lim_{x\rightarrow 0^+} f(x) =\infty$ it follows that the vertical asymptote is the line $x=0\text{.}$ Since $\displaystyle \lim_{x\rightarrow\pm\infty}f(x) = \lim_{x\rightarrow\pm\infty}\left( x-\frac{1}{x}\right)= \pm\infty$ we conclude that there is no horizontal asymptote. Finally, the fact $\displaystyle f(x) = x -\frac{1}{x}$ implies that $f$ has the slant (oblique) asymptote $y=x\text{.}$

3. For all $x\in \mathbb{R}\backslash \{ 0\}\text{,}$ $\displaystyle f'(x) = \frac{x^2+1}{x^2} >0$ so the function $f$ is increasing on $(-\infty ,0)$ and on $(0,\infty)\text{.}$ The function $f$ has no critical numbers and thus cannot have a local maximum or minimum.

4. Since $\displaystyle f''(x) = -\frac{2}{x^3}$ it follows that $f''(x)>0$ for $x\lt 0$ and $f''(x)\lt 0$ for $x>0\text{.}$ Therefore $f$ is concave upward on $(-\infty ,0)$ and concave downward on $(0,\infty)\text{.}$ There are no points of inflection.

###### 12.

Given $\ds f(x)=\frac{x^2}{x^2-1}\text{:}$

1. Find the domain of $f\text{.}$

2. Is $f$ an even function, odd function, or neither?

3. Find all the $x$- and $y$- intercepts of the graph of $f\text{.}$

4. Find all horizontal, vertical, and slant asymptotes of the graph of $f\text{.}$ If asymptote(s) of a certain kind are absent, explain why.

5. Find the intervals where the graph of $f$ is increasing or decreasing, and locations and values of the local maxima and minima.

6. Find the intervals where the the graph of $f$ is concave upward or downward and the inflection points.

7. Sketch the graph and clearly indicate the features found in parts (a)–(f).

Solution
1. All $x \in \mathbb{R}$ such that $x \neq \pm 1\text{.}$

2. Since $f(-x) = f(x) \text{,}$ $f$ is an even function.

3. $(0,0)\text{.}$

4. Vertical asymptotes when $x = \pm 1\text{.}$ Horizontal asymptote at $y=1\text{.}$

5. Increasing on $(0,-1) \cup (-1,-\infty) \text{.}$ Decreasing on $(0,1) \cup (1,\infty) \text{.}$

6. CCU on $(-\infty,-1) \cup(1, \infty) \text{.}$ CCD on $(-1,1)\text{.}$ No inflection points.

###### 13.

Given $\ds f(x)=\frac{x^2}{x^2+9}\text{:}$

1. Find the domain of $f\text{.}$

2. Is $f$ an even function, odd function, or neither?

3. Find all the $x$- and $y$- intercepts of the graph of $f\text{.}$

4. Find all horizontal, vertical, and slant asymptotes of the graph of $f\text{.}$ If asymptote of a certain kind are absent, explain why.

5. Find the intervals where the graph of $f$ is increasing or decreasing, and locations and values of the local maxima and minima.

6. Find the intervals where the the graph of $f$ is concave upward or downward and the inflection points.

7. Sketch the graph and clearly indicate the features found in parts (a)–(f).

Solution
1. All $x \in \mathbb{R}\text{.}$

2. Since $f(-x) = f(x) \text{,}$ $f$ is an even function.

3. $(0,0)\text{.}$

4. Horizontal asymptote at $y=1\text{.}$

5. Increasing on $(0,\infty) \text{.}$ Decreasing on $(-\infty,0)\text{.}$

6. CCU on $(-\sqrt{3},\sqrt{3})\text{.}$ CCD on $(-\infty,-\sqrt{3}) \cup (\sqrt{3},\infty)\text{.}$ Inflection points at $(-\sqrt{3}, 1/4)$ and $(\sqrt{3},1/4)\text{.}$

###### 14.

Suppose

\begin{equation*} f(x)=\frac{x}{x^2+4}, \, f^\prime(x)=\frac{4-x^2}{(x^2+4)^2}, \, f^{\prime\prime}(x)=\frac{2x^3-24x}{(x^2+4)^3}\text{.} \end{equation*}
1. Determine all vertical and horizontal asymptotes of $f\text{.}$

2. Determine the intervals on which the function is increasing or decreasing.

3. Determine all local maxima and minima of $f\text{.}$

4. Determine where $f$ is concave upward and where it is concave downward.

5. Find all inflection points.

6. Sketch the graph.

Solution
1. All $x \in \mathbb{R}\text{.}$

2. Increasing on $(-2,2)\text{.}$ Decreasing on $(-\infty,-2) \cup (2, \infty)\text{.}$

3. Local max at $(2, 1/4)\text{.}$ Local min at $(-2, -1/4) \text{.}$

4. CCU on $(-2\sqrt{3}, 0)\cup(2\sqrt{3},\infty)\text{.}$ CCD on $(-\infty, -2\sqrt{3}) \cup (0, 2\sqrt{3})\text{.}$

5. Inflection points at $(-2\sqrt{3}, -\sqrt{3}/8), (0,0), (2\sqrt{3}, \sqrt{3},8) \text{.}$

###### 15.

Suppose

\begin{equation*} f(x)=\frac{x^2}{x-1}, \, f^\prime(x)=\frac{x(x-2)}{(x-1)^2}, \, f^{\prime\prime}(x)=\frac{2}{(x-1)^3}\text{.} \end{equation*}
1. Determine the critical points of $f\text{.}$

2. Determine the intervals on which the function is increasing or decreasing. Classify the critical point(s) as either local maxima, local minima, or neither.

3. Determine where $f$ is concave upward and where it is concave downward. Identify any inflection points.

4. What is the end behaviour of $f$ (i.e. what is happening for large positive and negative $x$ values)? For instance, does $f$ have any horizontal or slant asymptotes?

5. Indicate which of the following graphs is the graph of $y=f(x)$ by circling the graph of your choice.

Top Left.
###### 16.

Consider the function $f(x)=x^3-2x^2-x+1$ on the interval $[-1,3]\text{.}$

1. The derivative of $f$ is:

2. The critical points for $f$ are:

3. The second derivative of $f$ is:

4. The points of inflection of $f$ are:

5. The intervals on which $f$ is increasing are:

6. The intervals on which $f$ is concave up are:

7. The intervals on which $f$ is concave down are:

8. $f$ has an absolute maximum at:

9. $f$ has an absolute minimum at:

10. $f$ has a local but not absolute maximum at:

11. $f$ has a local but not absolute minimum at:

12. Sketch the graph of $y=f(x)$ using all of the above information. All relevant points must be labeled.

###### 17.

The aim of this problem is to sketch the graph of $f(x)$ where

\begin{equation*} f(x)=\frac{x^2-2}{x^4}, \ f'(x)=-2\frac{x^2-4}{x^5}, \ f"(x)=2\frac{3x^2-20}{x^6}\text{.} \end{equation*}
1. Find the following limits and say what, if anything, they tell us about asymptotes of the graph of $y=f(x)\text{.}$

1. $\displaystyle \ds \lim _{x\to 0^-}f(x)$

2. $\displaystyle \ds \lim _{x\to 0^+}f(x)$

3. $\displaystyle \ds \lim _{x\to -\infty }f(x)$

4. $\displaystyle \ds \lim _{x\to \infty }f(x)$

2. Find the intervals on which $f$ is increasing and decreasing.

3. Find the intervals on which $f$ is concave up and concave down.

4. Find the $x$-intercepts.

5. Find the coordinates of all inflection points.

6. Indicate the coordinates of all points where the graph of $f$ has horizontal tangents. Are they local minima or maxima. Justify.

7. Sketch the graph of $y=f(x)$ using all of the above information. All relevant points must be labeled. You may need the numerical values: $\sqrt{2}\approx 1.414\text{,}$ $\ds \sqrt{\frac{20}{3}}\approx 2.582\text{,}$ and $\ds \frac{21}{200}=0.105\text{.}$

###### 18.

The goal of this exercise is to sketch the plot of $\ds f(x)=\frac{1}{(1+e^x)^2}\text{.}$

1. Find the domain of $f\text{.}$

2. Prove that the derivative of $f$ is given by $\ds f'(x)=-\frac{2e^x}{(1+e^x)^3}\text{.}$

3. Prove that the second derivative of $f$ is given by $\ds f"(x)=\frac{2e^x(2e^x-1)}{(1+e^x)^4}\text{.}$

4. Find the equations of the two horizontal asymptotes by finding limits of $f(x)$ at $x\to +\infty$ and $-\infty\text{.}$

5. Find any $x$- and $y$-intercepts.

6. Prove there are no critical points.

7. Prove that $\ds \left( -\ln 2,\frac{4}{9}\right)$ is the only inflection point.

8. Find the intervals on which $f$ is concave up and those on which $f$ is concave down.

9. Sketch the graph of $y=f(x)$ using all of the above information. (You may need $\ln 2\approx 0.7\text{,}$ $\frac{4}{9}\approx 0.44\text{.}$)

###### 19.

Let

\begin{equation*} f(x)=\frac{x^2-4x}{(x+4)^2}, \ f'(x)=\frac{4(3x-4)}{(x+4)^3}, \ f"(x)=-\frac{24(x-4)}{(x+4)^4}\text{.} \end{equation*}
1. Find any $x$- and $y$-intercepts.

2. Determine any horizontal asymptotes of $f(x)$ by taking appropriate limits.

3. Determine any vertical asymptotes of $f(x)$ by taking appropriate limits.

4. Fill in the sign of each factor of $f'(x)$ on the indicated intervals and thereby determine the sign of $f'(x)\text{.}$ Use this to determine where $f(x)$ is increasing/decreasing.

\begin{equation*} \begin{array}{|c|c|c|c|} \hline \amp (-\infty ,-4)\amp (-4,-4/3)\amp (4/3,\infty )\\ \hline (x+4)^3\amp \amp \amp \\ \hline 3x-4\amp \amp \amp \\ \hline f'(x)\amp \amp \amp \\ \hline f(x)\amp \amp \amp \\ \hline \end{array} \end{equation*}
5. Determine the location of any local extrema of $f(x)$ and indicate whether they are minima or maxima.

6. Fill in the sign of each factor of $f"(x)$ on the indicated intervals and thereby determine the sign of $f"(x)\text{.}$ Use this to determine where $f(x)$ is concave up/down.

\begin{equation*} \begin{array}{|c|c|c|c|} \hline \amp (-\infty ,-4)\amp (-4,-4)\amp (4,\infty )\\ \hline -24^3\amp \amp \amp \\ \hline (x+4)^4\amp \amp \amp \\ \hline x-4\amp \amp \amp \\ \hline f^{\prime\prime}(x)\amp \amp \amp \\ \hline f(x)\amp \amp \amp \\ \hline \end{array} \end{equation*}
7. Determine the locations of any inflection points of $f(x)\text{.}$

8. Sketch the graph of $y=f(x)$ using all of the above information.

###### 20.

Let

\begin{equation*} f(x)=\frac{4-4x}{x^2}, \ f^\prime(x)=\frac{4(x-2)}{x^3}, \ f^{\prime\prime}(x)=\frac{8(3-x)}{x ^4}\text{.} \end{equation*}

Determine the following. Show all your work.

1. The domain of $f\text{.}$

2. The $x$- and $y$-coordinates of all intercepts.

3. All asymptotes.

4. The intervals on which $f$ increases and the intervals on which $f$ decreases.

5. The $x$- and $y$-coordinates of all critical points, each classified as a local maximum, minimum or neither.

6. The intervals on which $f$ is concave up and the intervals on which $f$ is concave down.

7. The $x$- and $y$-coordinates of all inflection points.

8. Sketch the graph of $f$ using all of the above information and label all pertinent points and lines.

Solution
1. All $x \in \mathbb{R}$ such that $x \neq 0\text{.}$

2. $(1,0)\text{.}$

3. Vertical asymptote at $x=0\text{.}$ Horizontal asymptote at $y= 0\text{.}$

4. Increasing on $(-\infty,0) \cup (2,\infty)\text{.}$ Decreasing on $(0, 2)\text{.}$

5. One critical point at $(2,-1)\text{.}$ Local min.

6. CCU on $(-\infty, 0) \cup(0, 3)\text{.}$ CCD on $(3,\infty)\text{.}$

7. One inflection point at $(3, -8/9)\text{.}$

###### 21.

Sketch the graph of

\begin{equation*} y=4x^{1/3}+x^{4/3}\text{.} \end{equation*}

On your graph clearly indicate and label all intercepts, local extrema, and inflection points.

Solution

Note that the domain of the given function is the set of all real numbers. The $y$-intercept is the point $(0,0)$ and the $x$-intercepts are $( -4,0)$ and $(0,0)\text{.}$ From $\ds y'= \frac{4} {3}x^{1/3}\left( \frac{1}{x}+1\right)$ we conclude that $y'$ is not defined at $x=0$ and that $y'=0$ if $x=-1\text{.}$ Thus the critical numbers are $x=-1$ and $x=0\text{.}$ Also $y'\lt 0$ on $(-\infty,-1)$ and $y'>0$ on $(-1,0)\cup (0,\infty)\text{.}$ Hence the function has a local minimum at $x=-1\text{.}$ Note that the $y$-axis is a vertical asymptote to the graph of the given function. From $\ds y''= \frac{4}{9}x^{-5/3}(x-2)$ it follows that $y''(x)>0$ on $(-\infty,0)\cup (2,\infty$ and $y''(x)\lt 0$ on $(0,2)\text{.}$ Points of inflection are $(0,0)$ and $\ds (2,6\cdot 2^{1/3})\text{.}$

###### 22.

Consider the function $\ds f(x)=x^{2/3}\left( \frac{5}{2}-x\right)\text{.}$

1. Explain why $f$ is continuous for $x\in (-\infty ,\infty )\text{.}$

2. Determine the behavior of $f$ as $x\to \pm \infty\text{.}$

3. Given that $\ds f^\prime (x)=\frac{5}{3x^{1/3}}(1-x)\text{,}$ determine the regions of increase and decrease of $f\text{.}$

4. Identify the locations of the relative extrema of $f$ and classify them as maxima or minima.

5. Given that $\ds f^{\prime \prime} (x)=-\frac{5}{9x^{4/3}}(1+2x)\text{,}$ determine the concavity of $f\text{.}$

6. Identify the inflection points of $f\text{.}$

7. Sketch the graph of $f\text{.}$

###### 23.

Graph $y=x^x\text{.}$

###### 24.

Let

\begin{equation*} f(x)=\frac{x^2+2}{x^2-4}, \ f'(x)=\frac{-12x}{(x^2-4)^2}, \ f"(x)=\frac{12(3x^2+4)}{(x^2-4)^3}\text{.} \end{equation*}
1. Find the horizontal and vertical asymptotes of the given function (if any).

2. Find the intervals where the function is increasing or decreasing, and local maximum and local minimum values of the function (if any).

3. Find the intervals where the function is concave upward or downward and the inflection points.

4. Sketch a graph of the function.

###### 25.

Let

\begin{equation*} f(x)=x^2e^{-x}, \ f'(x)=(2x-x^2)e^{-x}, \ f"(x)=(2-4x+x^2)e^{-x}\text{.} \end{equation*}
1. Does the graph $f$ have any vertical or horizontal asymptotes. If so, what are they?

2. Determine the intervals of increase and decrease of this function. Also determine the extreme values.

3. Determine the intervals of upwards and downwards concavity of this function. Also determine any points of inflection.

4. Sketch a graph of this function. Clearly label the local extrema and points of inflection.

###### 26.

Let

\begin{equation*} f(x)=e^{1/x}, \ f'(x)=-\frac{e^{1/x}}{x^2}, \ f"(x)=\frac{e^{1/x}(2x+1)}{x^4}\text{.} \end{equation*}
1. What is the domain of $f\text{.}$

2. Determine any points of intersection of the graph of $f$ with the $x$ and $y$ axes.

3. Determine any horizontal asymptotes of $f\text{.}$

4. Determine any vertical asymptotes of $f\text{.}$

5. For each interval in the table below, indicate whether $f$ is increasing or decreasing.

\begin{equation*} \begin{array}{|c|c|c|} \hline \amp (-\infty ,0)\amp (0,\infty )\\ \hline f(x)\amp \amp \\ \hline \end{array} \end{equation*}
6. Determine the $x$ coordinates of any local extrema of $f\text{.}$

7. For each interval in the table below, indicate whether $f$ is concave up or concave down.

\begin{equation*} \begin{array}{|c|c|c|c|} \hline \amp (-\infty ,-1/2)\amp (-1/2,0)\amp (0,\infty )\\ \hline f(x)\amp \amp \amp \\ \hline \end{array} \end{equation*}
8. Determine the $x$ coordinates of any inflection points on the graph of $f\text{.}$

9. Sketch the graph of $y=f(x)$ using all of the above information. All relevant points must be labeled.

###### 27.

Let $\ds f(x)=e^{-2x^2}\text{.}$

1. Find any horizontal and vertical asymptotes of $f(x)\text{.}$

2. Find the intervals where $f(x)$ is increasing and the intervals where $f(x)$ is decreasing.

3. Find the $x$ values at which any local maxima or local minima occur and state whether a local maximum or a local minimum occurs.

4. Give the interval where $f(x)$ is concave up and the intervals where $f(x)$ is concave down.

5. Find the inflection points of $f(x)\text{.}$

1. $y=0\text{.}$

2. Increasing on $(-\infty ,0)\text{.}$

3. Local maximum at $x=0\text{.}$

4. Concave down on $(-2,2)\text{.}$

5. Inflection points at $x=\pm 2\text{.}$

###### 28.

Consider the function $\ds f(x)=x^3e^{-x+5}\text{.}$ The first and second derivatives are:

\begin{equation*} f^\prime(x) =x^2(3-x)e^{-x+5}, \ f^{\prime \prime(x)}=x(x^2-6x+6)e^{-x+5}\text{.} \end{equation*}
1. Determine the critical points of $f\text{.}$

2. Determine the intervals on which the function is increasing, and those on which the function is decreasing. Classify the critical point(s) as either local maxima, local minima, or neither.

3. Determine where $f$ is concave up and where it is concave down. Identify any inflection points.

4. What is the end behaviour of $f$ (i.e. what is happening as $x\to \infty$ and $x\to -\infty$)?

5. Indicate which of graphs in the Figure below is the graph of $y=f(x)\text{.}$ Also, identify the critical points and inflection points on the graph you've chosen and write in the $x$-coordinates of these points.

1. $(0,0)\text{,}$ $(3,9e^2)\text{.}$

2. Increasing on $(-\infty ,3)$ and decreasing on $(3,\infty )\text{.}$ A local (global) maximum at $(3,9e^2)\text{.}$ The other critical point is neither a local maximum nor a local minimum.

3. Note that $x^2-6x+6=(x-(3-\sqrt{3}))(x-(3+\sqrt{3}))\text{.}$ The function is concave up on $(0,3-\sqrt{3})$ and $(3+\sqrt{3},\infty)$ and concave down on $(-\infty ,0)$ and $(3-\sqrt{3},3+\sqrt{3})\text{.}$ The inflection points are $(0,0)\text{,}$ $(3-\sqrt{3}, (3-\sqrt{3})^3e^{2+\sqrt{3}})\text{,}$ and $(3+\sqrt{3},(3+\sqrt{3})^3e^{2-\sqrt{3}})\text{.}$

4. $\ds \lim _{x\to -\infty }f(x)=-\infty\text{,}$ $\ds \lim _{x\to \infty }f(x)=0\text{.}$

5. Top Left.

###### 29.

Let $\ds f(x)=\frac{18(x-1)}{x^2}\text{.}$ Then $\ds f'(x)=\frac{18(2-x)}{x^3}$ and $\ds f"(x)=\frac{36(x-3)}{x^4}\text{.}$ Give the following:

1. The domain of $f\text{.}$

2. The $x$ and $y$ coordinates of all intercepts of $f\text{.}$

3. The equations of asymptotes.

4. The intervals on which $f$ is increasing and those on which $f$ is decreasing.

5. The $x$ and $y$ coordinates of all critical points, each classified as a max, min, or neither.

6. The intervals on which $f$ is concave up and those on which $f$ is concave down.

7. The $x$ and $y$ coordinates of all inflection points of $f\text{.}$

8. Sketch the graph of $y=f(x)$ using all of the above information. All relevant points must be labeled.

###### 30.

Let $\ds f(x)=\frac{x+3}{\sqrt{x^2+1}}\text{.}$ Then $\ds f'(x)=\frac{1-3x}{(x^2+1)^{3/2}}$ and $\ds f"(x)=\frac{6x^2-3x-3}{(x^2+1)^{5/2}}\text{.}$ Give the following:

1. The domain of $f\text{.}$

2. The $x$ and $y$ coordinates of all intercepts of $f\text{.}$

3. The equations of asymptotes, if any.

4. All critical points, if any.

5. The intervals on which $f$ is increasing and those on which $f$ is decreasing.

6. The classification of each critical point, if any, as a minimum or maximum, local or global, or not an extremum.

7. The intervals on which $f$ is concave up and those on which $f$ is concave down.

8. The $x$ and $y$ coordinates of all inflection points of $f\text{,}$ if any.

9. Sketch the graph of $y=f(x)$ using all of the above information. All relevant points must be labeled.

###### 31.

Consider the function

\begin{equation*} f(x)=(5-2x)x^{2/3}\text{.} \end{equation*}

For your convenience, the first and second derivative of $f(x)$ are

\begin{equation*} f'(x)=\frac{10(1-x)}{3x^{1/3}}, \ f"(x)=-\frac{10(1+2x)}{9x^{4/3}}\text{.} \end{equation*}
1. Locate any intercepts of $f\text{.}$

2. Locate any asymptotes of $f\text{.}$

3. Determine where $f$ is increasing and decreasing.

4. Locate the local extrema of $f\text{.}$

5. Determine where $f$ is concave upward or concave downward.

6. Locate all inflection points of $f\text{.}$

7. Sketch a graph of the function.

###### 32.

Consider the function

\begin{equation*} f(x)=\frac{x}{x^2-1}\text{.} \end{equation*}

For your convenience, the first and second derivative of $f(x)$ are

\begin{equation*} f'(x)=-\frac{x^2+1}{(x^2-1)^2}, \mbox{ and } f"(x)=\frac{2x(x^2+3)}{(x^2-1)^3}\text{.} \end{equation*}
1. Determine any horizontal and vertical asymptotes of $f\text{.}$

2. Determine the open intervals on which $f$ is increasing as well those on which $f$ is decreasing.

3. Determine the open intervals on which $f$ is concave upward as well those on which $f$ is concave downward.

4. Based on the information found in Parts (a) to (c), sketch a graph of $f\text{.}$ Indicate any relative extremum, inflection points and intercepts on your graph.

###### 33.

Given is the function

\begin{equation*} f(x)=\frac{x^3-2x}{3x^2-9}\text{.} \end{equation*}
1. Give the domain of $f\text{.}$

2. Is $f$ an even function, odd function, or neither?

3. Find all the $x$ and $y$-intercepts of the graph $f\text{.}$

4. Find all horizontal, vertical, and slant asymptotes of the graph of $f\text{.}$ If asymptotes of a certain kind are absent, explain why.

5. Find the intervals where the graph of $f$ is increasing or decreasing and the locations and values of the local maxima and minima.

6. Find the intervals where the graph of $f$ is concave upward and concave downward and the infection points.

7. Sketch the graph and clearly indicate the features found in parts (a)-(f).

###### 34.

Suppose $f$ is a function satisfying the following conditions:

\begin{equation*} \lim _{x\to 3^+}f(x)=+\infty ,\ \lim _{x\to 3^-}f(x)=-\infty ,\ \lim _{x\to \infty }f(x)=2 ,\ \lim _{x\to -\infty }f(x)=-1\text{,} \end{equation*}

$f(0)=-3\text{,}$ and

\begin{equation*} f'(x)\lt 0 \mbox{ for all } x\not= 3, \ f"(x)>0 \mbox{ for all } x> 3, \ f"(x)\lt 0 \mbox{ for all } x\lt 3\text{.} \end{equation*}

Draw a graph of the function $f$ with all asymptotes and intercepts clearly labeled.

Solution

It is given that the $y$-intercept is the point $(0,-3)\text{.}$ Note that the given function has a vertical asymptote $x=3$ and two horizontal asymptotes, $y=-1\text{,}$ when $x\to -\infty\text{,}$ and $y=2\text{,}$ when $x\to \infty\text{.}$ Also, the function $f$ is decreasing on $(-\infty ,3)$ and $(3,\infty )\text{.}$ Finally, $f$ is concave upwards on $(3,\infty )$ and concave downwards on $(-\infty ,3)\text{.}$

###### 35.
1. Plot the graph of a function $f$ which has only one point of discontinuity on its domain $[-4,\infty )$ and that satisfies:

\begin{equation*} \ds \begin{array}{lllll} \ds \lim _{x\to \infty }f(x)=-2\amp \amp f^{\prime\prime}(x)\lt 0\amp \mbox{if} \amp -4\lt x\lt -1\\ \ds \lim _{x\to 0^- }f(x)=\infty \amp \amp f^{\prime\prime}(x)>0\amp \mbox{if} \amp -1\lt x\lt 0\\ f(0)=2\amp \amp f"(x)\lt 0\amp \mbox{if} \amp 0\lt x\lt 4\\ \amp \amp f^{\prime\prime}(x)>0\amp \mbox{if} \amp 4\lt x\lt \infty \\ \amp \amp f^\prime(x)\lt 0\amp \mbox{if} \amp -4\lt x\lt -1 \\ \amp \amp f^\prime(x)>0\amp \mbox{if} \amp -1\lt x\lt 0 \\ \amp \amp f^\prime(x)>0\amp \mbox{if} \amp 0\lt x\lt 2 \\ \amp \amp f^\prime(x)\lt 0\amp \mbox{if} \amp 2\lt x\lt \infty \end{array} \end{equation*}
2. Find all points of inflection for this graph, and for each point of inflection, determine if it is possible that $f"(x)=0\text{.}$

Solution
1. The graph has a vertical asymptote $y=0$ and a horizontal asymptote $x=-2\text{.}$ The following table summarizes the rest of the given information.

\begin{equation*} \begin{array}{c||c|c|c|c|c|} \text{Interval} & (-4,-1) & (-1,0) & (0,2) & (2,4) & (4,\infty) \\ \hline \hline \text{Monotonity} & \text{Decreasing} & \text{Increasing} & \text{Increasing}& \text{Decreasing} & \text{Decreasing}\\ \hline \text{Concavity} & \text{Downwards} & \text{Upwards} & \text{Downwards}& \text{Downwards} & \text{Upwards}\\ \hline \end{array} \end{equation*}

For example:

2. There are two points of inflection, $x=-1$ and $x=4\text{.}$ We note that $x=-1$ is also a critical number and that by the first derivative test there is a local minimum at $x=-1\text{.}$ If $f"(-1)=0\text{,}$ then $f'(-1)$ exists and $f^\prime(-1)=0\text{.}$ This would imply that at this point the graph of $f$ is above the tangent line at $x=-1$ which contradicts the fact that the curve crosses its tangent line at each inflection point. It follows that $f'(-1)$ does not exist and therefore $f"(-1)$ does not exist.

###### 36.

The graphs of $r\text{,}$ $s\text{,}$ $f^\prime\text{,}$ and $g^\prime$ are shown below, as labelled (these functions are all unrelated). For each question, tick the box if the corresponding function $r\text{,}$ $s\text{,}$ $f\text{,}$ or $g$ has the stated property. Note that you can tick more than one box.

1. The function is increasing over the interval $(1,3)\text{.}$

\begin{equation*} \begin{array}{ccccc} r \ \square \ \amp s \ \square \ \amp f \ \square \ \amp g \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
2. The function has a critical point when $x=3\text{.}$

\begin{equation*} \begin{array}{ccccc} r \ \square \ \amp s \ \square \ \amp f \ \square \ \amp g \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
3. The function has an inflection point when $x=1\text{.}$

\begin{equation*} \begin{array}{ccccc} r \ \square \ \amp s \ \square \ \amp f \ \square \ \amp g \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
4. The function is concave up over the interval $(0,2)\text{.}$

\begin{equation*} \begin{array}{ccccc} r \ \square \ \amp s \ \square \ \amp f \ \square \ \amp g \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
5. The function has a point where the second derivative does not exist.

\begin{equation*} \begin{array}{ccccc} r \ \square \ \amp s \ \square \ \amp f \ \square \ \amp g \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
1. $\surd\text{:}$ $r\text{,}$ $s\text{,}$ $g\text{.}$

2. $\surd\text{:}$ $r\text{,}$ $s\text{,}$ $f\text{,}$ $g\text{.}$

3. $\surd\text{:}$ $s\text{,}$ $f\text{.}$

4. $\surd\text{:}$ $r\text{,}$$g\text{.}$

5. $\surd\text{:}$ $r\text{.}$

###### 37.

The graphs of four functions are shown below. For each question, tick the box if the corresponding function has the stated property. Note that you can tick more than one box.

1. The derivative of the function is zero at $x=-1\text{.}$

\begin{equation*} \begin{array}{ccccc} A \ \square \ \amp B \ \square \ \amp C \ \square \ \amp D \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
2. The function has a point where the second derivative does not exist.

\begin{equation*} \begin{array}{ccccc} A \ \square \ \amp B \ \square \ \amp C \ \square \ \amp D \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
3. The derivative of the function is negative on the whole interval $(-2,0)\text{.}$

\begin{equation*} \begin{array}{ccccc} A \ \square \ \amp B \ \square \ \amp C \ \square \ \amp D \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
4. The function has a critical point when $x=3\text{.}$

\begin{equation*} \begin{array}{ccccc} A \ \square \ \amp B \ \square \ \amp C \ \square \ \amp D \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
5. The second derivative is positive over the whole interval $(2,5)\text{.}$

\begin{equation*} \begin{array}{ccccc} A \ \square \ \amp B \ \square \ \amp C \ \square \ \amp D \ \square \ \amp \mbox{None of them} \ \square \end{array} \end{equation*}
1. $\surd\text{:}$ C, D.

2. $\surd\text{:}$ A.

3. $\surd\text{:}$ A, D.

4. $\surd\text{:}$ A, B, C, D.

5. $\surd\text{:}$ B.

###### 38.

For what values of the constants $a$ and $b$ is $(1,6)$ a point of inflection of the curve $y=x^3+ax^2+bx+1\text{?}$ Justify your answer.

Hint
Solve the system $y(1)=a+b+2=6$ and $y^{\prime\prime}(1)=6+2a=0\text{.}$
$a=-3\text{,}$ $b=7\text{.}$