## Section1.4The Mean Value Theorem

###### Motivating Questions

How can we use the size of the derivative to quantify how fast a function is increasing or decreasing?

In Active Calculus 3.1 we learned about critical points and how to use them to find relative maximums and relative minimums. Without giving it a name, we used the following theorem.

In Active Calculus 3.3 we learned the following theorem, which we state now in slightly different form.

###### Preview Activity1.4.1

For each of the following cases, either draw a function that satisfies the conditions or explain why it is impossible.

1. $f'(c)=0$ and $f$ has a relative minimum at $c\text{.}$
2. $f'(c)=1$ and $f$ has a relative minimum at $c\text{.}$
3. $f'(c)$ does not exist and $f$ has a relative minimum at $c\text{.}$
4. $f'(c)=0$ and $f$ does not have a relative maximum or relative minimum at $c\text{.}$
5. $f$ is not continuous on $[a,b]$ and $f$ has an absolute maximum at $c=(a+b)/2\text{.}$
6. $f$ attains an absolute maximum on $[a,b]$ but does not attain an absolute minimum on $[a,b]\text{.}$

We first consider a Theorem that shows that if a differentiable function had zero net change on an interval, then somewhere it has a horizontal tangent.

Since $f$ is continuous on a closed interval $[a,b]\text{,}$ the Extreme Value Theorem 1.4.2 applies, so $f$ attains both an absolute maximum and an absolute minimum on $[a,b]\text{.}$ We then split into three cases:

• $f(a)$ is both an absolute maximum and an absolute minimum.

• Thus for any $x$ in $(a,b)\text{,}$ $f(x)$ cannot be bigger or smaller than $f(a)\text{,}$ so $f$ is the constant function $f(x)=f(a)\text{.}$
• Differentiating gives $f'(x)=0\text{,}$ so every $c$ in $(a,b)$ satisfies $f'(c)=0\text{.}$
• $f(a)$ is not an absolute maximum.

• Since $f(a)=f(b)\text{,}$ $f(b)$ is also not an absolute maximum.
• Therefore the absolute maximum is $f(c)$ for some $c$ in $(a,b)\text{.}$
• Since $c$ is not at an endpoint, $f$ also has a relative maximum at $c\text{.}$
• By Fermat's Theorem 1.4.1, either $f'(c)=0$ or $f'(c)$ does not exist.
• Since we assumed $f$ is differentible on $(a,b)\text{,}$ we must have $f'(c)=0\text{.}$
• $f(a)$ is not an absolute minimum. Use the same logic as the case where $f(a)$ is not an absolute maximum.
###### Activity1.4.2

For each of the following cases, either draw a function that satisfies the conditions or explain why it is impossible.

1. $f$ satisfies each of the assumptions of Rolle's Theorem 1.4.3, $f$ attains its absolute maximum at $a\text{,}$ and $f$ attains its absolute minimum at $a\text{.}$
2. $f$ satisfies each of the assumptions of Rolle's Theorem 1.4.3 and $f$ does not attain its absolute maximum at $a\text{.}$
3. $f$ satisfies each of the assumptions of Rolle's Theorem 1.4.3 and $f$ does not attain its absolute minimum at $a\text{.}$
###### Activity1.4.3

Consider the function $f(x)=x^5+x+3\text{.}$

1. Use the Intermediate Value Theorem 1.2.1 to show $f$ has at least one root.
2. Use Rolle's Theorem 1.4.3 to show that if $f$ has more than one root, then $f'$ has at least one root.
3. Show that $f'(x) \ge 1\text{.}$
4. How many roots does $f$ have?

Now we turn Rolle's Theorem 1.4.3 on a slant and consider a Theorem that shows somewhere the function has a tangent line with slope the same as its mean rate of change.

Let

\begin{equation*} g(x)=f(x) -\frac{f(b)-f(a)}{b-a}(x-a)\, \end{equation*}

for which we can compute

\begin{equation*} g'(x)=f'(x) -\frac{f(b)-f(a)}{b-a}\,. \end{equation*}

Since $f$ is continuous on $[a,b]\text{,}$ so is $g\text{.}$ Since $f$ is differentiable on $(a,b)\text{,}$ so is $g\text{.}$ We can check directly that $g(a)=g(b)=f(a)\text{.}$ Thus Rolle's Theorem 1.4.3 applies to $g\text{,}$ and there is $c$ in $(a,b)$ that satisfies $g'(c)=0\text{.}$ Plugging into $g'(x)$ gives

\begin{equation*} 0=g'(c)=f'(c) -\frac{f(b)-f(a)}{b-a} \end{equation*}

so

\begin{equation*} f'(c)=\frac{f(b)-f(a)}{b-a}\text{.} \end{equation*}
###### Activity1.4.4
1. Draw a function $f$ that satisfies each of the hypotheses of the Mean Value Theorem 1.4.4 and has $f(a)\not = f(b)\text{.}$
2. Draw the secant line connecting $(a,f(a))$ and $(b,f(b))\text{.}$ Compute its slope in terms of $a\text{,}$ $b\text{,}$ $f(a)\text{,}$ and $f(b)\text{.}$
3. Find a point on your graph of $f$ where the tangent line is parallel to this secant line.
###### Activity1.4.5

Suppose we know $f$ is a differentiable function with $f(1)=5$ and $f'(x) \ge 2\text{.}$

1. Sketch several such functions on the interval $[0,4]$
2. What is the greatest possible value for $f(0)\text{?}$
3. What is the greatest possible value for $f(4)\text{?}$
4. What is the least possible value for $f(0)\text{?}$
5. What is the least possible value for $f(4)\text{?}$

### SubsectionExercises

###### 1

Consider the function $f(x)=x^3-4x$ on the interval $[-2,2]\text{.}$

1. Verify that this function satisfies each of the hypotheses of Rolle's Theorem 1.4.3 on this interval.
2. Find all numbers $c$ that satisfy the conclusion of Rolle's Theorem 1.4.3.
###### 2

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

1. Verify that this function satisfies each of the hypotheses of the Mean Value Theorem 1.4.4 on this interval.
2. Find all numbers $c$ that satisfy the conclusion of the Mean Value Theorem 1.4.4.
###### 3

If $f$ is a differentiable function with $f(0)=2$ and $f'(x) \le 4\text{,}$ then what is the maximum value $f(3)$ could be? What is the minimum value $f(3)$ could be?.