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Section 3.4 Mean Value Theorem

Use the Mean Value Theorem to solve the following problems.

  1. Verify that the function

    \begin{equation*} g(x)=\frac{3x}{x+7} \end{equation*}

    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.

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

  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.

    1. 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 \(f(b)-f(a)=\) . . . . . . . . . . .

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

    3. 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 \((0,2)\text{.}\)