Question 1: Straightforward application of Rolle's theorem

Consider the function $f(x) = 9 - (x-3)^2$ on the interval $[0, 6]$. (Note that $f(0) = f(6) = 0$.) Find the value(s) of $c$ that satisfy Rolle's Theorem.

Question 2: Straightforward application of MVT

Consider the function $f(x) = x^2$ on the interval $[1, 4]$. Find the value(s) of $c$ that satisfy the Mean-Value Theorem.

Question 3: How large can *f(b)* be? (based on an actual university exam question)

Suppose that $f(1) = 2$, and that $f'(x) \le 3$ for all values of $x$. How large can $f(5)$ possibly be?

[*Hint:* Use the Mean Value Theorem.]

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Question 4: Prove if *f'(x)* > 0, then *f(x)* is an increasing function

Use the Mean Value Theorem to prove that if $f(x)$ is differentiable and $f'(x) > 0$ for all $x$, then $f(x)$ is an increasing function.

Question 5: Prove if *f'(x)* = 0, then ...; if *f'(x)* = *g'(x)*, then ...

Prove the following.**(a)** Use the Mean Value Theorem to prove that if $f'(x) = 0$ for all $x$ in an interval $(a,b)$, then $f$ is constant on $(a,b)$.**(b)** Use the Mean Value Theorem to prove that if $f'(x) = g'(x)$ for all $x$ in an interval $(a,b)$, then $f'(x) = g'(c) + C$ on $(a,b)$, where $C$ is some constant.

[*Hint: *Consider the function $h(x) = f(x) - g(x)$, and use the result of part (a).]

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Question 6: Prove that |sin(a) - sin(b)| ≤ |b-a|

Use the Mean Value Theorem to prove that $|\sin (a) - \sin (b)| \le |b-a|$ for all real numbers $a$ and $b$.

Question 7: Prove that |cos(2a) - cos(2b)| ≤ 2|b-a|

Use the Mean Value Theorem to prove that $|\cos (2a) - \cos (2b)| \le 2 |b-a|$ for all real numbers $a$ and $b$.

Question 8: Prove that *e^x* > 1 + *x* for *x* > 0 (based on an actual university exam question)

Use the Mean Value Theorem to prove that $e^x > 1 + x$ for all $x > 0$.

Question 9: Prove that only one root (based on an actual university exam question)

Show that $f(x) = x^5 + 6x^3 + x - 4$ has exactly one real root.

Question 10: Prove that only one root (based on an actual university exam question)

Show that $g(x) = 6x - \cos x$ has exactly one real root.

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Mean Value Theorem (MVT):If $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there is a number $c$ in $(a,b)$ such that

$$f'(c) = \frac{f(b) – f(a)}{b-a}$$

or, equivalently,

$$f(b) – f(a) = f'(c)(b-a)$$

In words, there is at least one value $c$ between $a$ and $b$ where the tangent line is parallel to the secant line that connects the interval’s endpoints. (See the figures.)

[Click on any figure to see a larger version.]

Rolle’s Theorem:In Calculus texts and lecture, Rolle’s theorem is given first since it’s used as part of the proof for the Mean Value Theorem (MVT). You can easily remember it, though, as just a special case of the MVT: it has the same requirements about continuity on $[a,b]$ and differentiability on $(a,b)$, and the additional requirement that $f(a) = f(b)$. In that case, the MVT says that

\begin{align*}

f(b) – f(a) &= f'(c)(b-a) \\

0 &= f'(c)(b-a) \\

f'(c) &= 0 \text{ for some number $c$ in the open interval $(a,b)$}

\end{align*}

See the figure.

The Mean-Value Theorem (and to a lesser extent) Rolle’s Theorem often appear on exams as questions that ask you to prove something or another. The problems below focus on such questions.