if $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ then there exists $c \in (a, b)$ such that $$f' (c) = \frac{f (b) - f (a)}{b - a}$$

if $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ and if $f (a) = f (b),$ then there exists $c \in (a, b)$ such that $f' (c) = 0$

Calculus Volume 1 · 4.4 The Mean Value Theorem

The Mean Value Theorem: practice problems

26 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 149
Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.
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Exercises 151
If you have a function with a discontinuity, is it still possible to have $f^{'} (c) (b - a) = f (b) - f (a)?$ Draw such an example or prove why not.
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Exercises 153
For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.
$y = \frac{1}{x ^{3}}$
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Exercises 155
For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.
$y = x^{2} - 4$
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Exercises 157
For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points $c$ such that $f^{'} (c) (b - a) = f (b) - f (a) .$
[T] $y = 3 x^{3} + 2 x + 1$ over
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Exercises 159
For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points $c$ such that $f^{'} (c) (b - a) = f (b) - f (a) .$
[T] $y = x^{2} cos (π x)$ over
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Exercises 161
For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) - f (0) = f^{'} (c) (2 - 0) .$
$f (x) = x^{3}$
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Exercises 163
For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) - f (0) = f^{'} (c) (2 - 0) .$
$f (x) = cos (2 π x)$
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Exercises 165
For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) - f (0) = f^{'} (c) (2 - 0) .$
$f (x) = (x - 1)^{10}$
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Exercises 167
For the following exercises, show there is no $c$ such that $f (1) - f (- 1) = f^{'} (c) (2) .$ Explain why the Mean Value Theorem does not apply over the interval $[- 1, 1] .$
$f (x) = ∣ x - \frac{1}{2} ∣$
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Exercises 169
For the following exercises, show there is no $c$ such that $f (1) - f (- 1) = f^{'} (c) (2) .$ Explain why the Mean Value Theorem does not apply over the interval $[- 1, 1] .$
$f (x) = ∣ x ∣$
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Exercises 171
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.
$y = e^{x}$ over $[0, 1]$
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Exercises 173
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.
$f (x) = tan (2 π x)$ over $[0, 2]$
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Exercises 175
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.
$y = \frac{1}{∣ x + 1∣}$ over $[0, 3]$
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Exercises 177
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.
$y = \frac{x ^{2} + 3 x + 2}{x}$ over $[- 1, 1]$
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Exercises 179
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.
$y = ln (x + 1)$ over $[0, e - 1]$
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Exercises 181
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.
$y = 5 + ∣ x ∣$ over $[- 1, 1]$
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Exercises 183
For the following exercises, consider the roots of the equation.
Find the conditions for exactly one root (double root) for the equation $y = x^{2} + b x + c$
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Exercises 185
For the following exercises, use a calculator to graph the function over the interval $[a, b]$ and graph the secant line from $a$ to $b .$ Use the calculator to estimate all values of $c$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $c,$ if possible, or write the final equation and use a calculator to estimate to four digits.
[T] $y = tan (π x)$ over $[- \frac{1}{4}, \frac{1}{4}]$
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Exercises 187
For the following exercises, use a calculator to graph the function over the interval $[a, b]$ and graph the secant line from $a$ to $b .$ Use the calculator to estimate all values of $c$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $c,$ if possible, or write the final equation and use a calculator to estimate to four digits.
[T] $y = ∣ x^{2} + 2 x - 4∣$ over
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Exercises 189
For the following exercises, use a calculator to graph the function over the interval $[a, b]$ and graph the secant line from $a$ to $b .$ Use the calculator to estimate all values of $c$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $c,$ if possible, or write the final equation and use a calculator to estimate to four digits.
[T] $y = x + 1 + \frac{1}{x ^{2}}$ over
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Exercises 191
For the following exercises, use a calculator to graph the function over the interval $[a, b]$ and graph the secant line from $a$ to $b .$ Use the calculator to estimate all values of $c$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $c,$ if possible, or write the final equation and use a calculator to estimate to four digits.
Two cars drive from one stoplight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.
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Exercises 193
For the following exercises, use a calculator to graph the function over the interval $[a, b]$ and graph the secant line from $a$ to $b .$ Use the calculator to estimate all values of $c$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $c,$ if possible, or write the final equation and use a calculator to estimate to four digits.
Show that $y = csc^{2} x$ and $y = cot^{2} x$ have the same derivative. What can you say about
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Review Exercises 525
True or False? Justify your answer with a proof or a counterexample. Assume that $f (x)$ is continuous and differentiable unless stated otherwise.
If $f (- 1) = - 6$ and $f (1) = 2,$ then there exists at least one point $x \in [- 1, 1]$ such that $f^{'} (x) = 4.$
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Review Exercises 527
True or False? Justify your answer with a proof or a counterexample. Assume that $f (x)$ is continuous and differentiable unless stated otherwise.
There is a function such that $f (x) < 0, f^{'} (x) > 0,$ and $f^{''} (x) < 0.$ (A graphical “proof” is acceptable for this answer.)
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Review Exercises 529
True or False? Justify your answer with a proof or a counterexample. Assume that $f (x)$ is continuous and differentiable unless stated otherwise.
Given the graph of $f^{'},$ determine where $f$ is increasing or decreasing.
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Key terms · 2

mean value theorem: if $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ then there exists $c \in (a, b)$ such that $$f' (c) = \frac{f (b) - f (a)}{b - a}$$
rolle’s theorem: if $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ and if $f (a) = f (b),$ then there exists $c \in (a, b)$ such that $f' (c) = 0$

Learn 4.4 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 4.4 in the textbook · This section on openstax.org

The Mean Value Theorem: practice problems

26 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 149

Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.

Exercises 151

If you have a function with a discontinuity, is it still possible to have $f^{'} (c) (b - a) = f (b) - f (a)?$ Draw such an example or prove why not.

Exercises 153

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

$y = \frac{1}{x ^{3}}$

Exercises 155

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

$y = x^{2} - 4$

Exercises 157

For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points $c$ such that $f^{'} (c) (b - a) = f (b) - f (a) .$

[T] $y = 3 x^{3} + 2 x + 1$ over

Exercises 159

[T] $y = x^{2} cos (π x)$ over

Exercises 161

For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) - f (0) = f^{'} (c) (2 - 0) .$

$f (x) = x^{3}$

Exercises 163

For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) - f (0) = f^{'} (c) (2 - 0) .$

$f (x) = cos (2 π x)$

Exercises 165

For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) - f (0) = f^{'} (c) (2 - 0) .$

$f (x) = (x - 1)^{10}$

Exercises 167

For the following exercises, show there is no $c$ such that $f (1) - f (- 1) = f^{'} (c) (2) .$ Explain why the Mean Value Theorem does not apply over the interval $[- 1, 1] .$

$f (x) = ∣ x - \frac{1}{2} ∣$

Exercises 169

For the following exercises, show there is no $c$ such that $f (1) - f (- 1) = f^{'} (c) (2) .$ Explain why the Mean Value Theorem does not apply over the interval $[- 1, 1] .$

$f (x) = ∣ x ∣$

Exercises 171