## Exercises 3.4 Mean Value Theorem

Use the Mean Value Theorem to solve the following problems.

###### 1.

Verify that the function

satisfies the hypothesis of the Mean Value Theorem on the interval \([-1,2]\text{.}\) Then find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. Leave your final answer(s) exact.

Since \(x+7\not= 0\) for all \(x\in [-1,2]\) it follows that the function \(g\text{,}\) as a rational function, is continuous on the closed interval \([-1,2]\) and differentiable on the open interval \((-1,2)\text{.}\) Therefore the function \(g\) satisfies he hypothesis of the Mean Value Theorem on the interval \([-1,2]\text{.}\) By the Mean Value Theorem there is \(c\in (-1,2)\) such that \(\ds g'(c)=\frac{g(2)-g(-1)}{2-(-1)}\text{.}\) Thus the question is to solve \(\ds \frac{21}{(c+7)^2}=\frac{7}{18}\) for \(c\text{.}\) Hence \(\ds c=-7\pm 3\sqrt{6}\text{.}\) Clearly \(-7-3\sqrt{6}\lt -1\) and this value is rejected. From

it follows that \(c=-7+3\sqrt{6}\in(-1,2)\) and it is the only value that satisfies the conclusion of the Mean Value Theorem.

###### 2.

Use the Mean Value Theorem to show that \(|\sin b-\sin a|\leq |b-a|\text{,}\) for all real numbers \(a\) and \(b\text{.}\)

The inequality is obviously satisfied if \(a=b\text{.}\) Let \(a,b\in \mathbb{R}\text{,}\) \(a\lt b\text{,}\) and let \(f(x)=\sin x\text{,}\) \(x\in [a,b]\text{.}\) Clearly the function \(f\) is continuous on the closed interval \([a,b]\) and differentiable on \((a,b)\text{.}\) Thus, by the Mean value Theorem, there is \(c\in (a,b)\) such that \(\ds \cos c=\frac{\sin b-\sin a}{b-a}\text{.}\) Since \(|\cos c|\leq 1\) for all real numbers \(c\) it follows that \(|\sin b-\sin a|\leq |b-a|\text{.}\)

###### 3.

Two horses start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed.

Let \(f(t)\) be the distance that the first horse covers from the start in time \(t\) and let \(g(t)\) be the distance that the second horse covers from the start in time \(t\text{.}\) Let \(T\) be time in which the two horses finish the race. It is given that \(f(0)=g(0)\) and \(f(T)=g(T)\text{.}\) Let \(F(t)=f(t)-g(t)\text{,}\) \(t\in [0,T]\text{.}\) As the difference of two position functions, the function \(F\) is continuous on the closed interval \([0,T]\) and differentiable on the open interval \((0,T)\text{.}\) By the Mean value Theorem there is \(c\in (0,T)\) such that \(\ds F'(c)=\frac{F(T)-F(0)}{T-0}=0\text{.}\) It follows that \(f'(c)=g'(c)\) which is the same as to say that at the instant \(c\) the two horse have the same speed. (Note: It is also possible to use Rolle's theorem.)

###### 4.

Complete the following statement of the Mean Value Theorem precisely. Let \(f\) be a function that is continuous on the interval . . . . . . . and differentiable on the interval . . . . . . . Then there is a number \(c\) in \((a,b)\) such that \(f(b)-f(a)=\) . . . . . . . . . . .

###### 5.

Suppose that \(-1\leq f^\prime (x)\leq 3\) for all \(x\text{.}\) Find similar lower and upper bounds for the expression \(f(5)-f(3)\text{.}\)

###### 6.

Suppose \(g(x)\) is a function that is differentiable for all \(x\text{.}\) Let \(h(x)\) be a new function defined by \(h(x)=g(x)+g(2-x)\text{.}\) Prove that \(h^\prime (x)\) has a root in the interval \((0,2)\text{.}\)