Mean Value Theorem Practice Problems

Problem: Mean Value Theorem #1

Find all values of $c$ that satisfy the mean value theorem for the function:

$\Large{f(x) = x^2-4x+4}$

On the interval $[0,3]$

Solution

First, since the given function is a polynomial, we know that it is continuous and differentiable on the given interval. This means that the mean value theorem applies.

Step 1:

First, let’s find the slope of the line which connects the two endpoints of the interval, which are given as a = 0 and b = 3. So,

\large{m_{ab} = \frac{f(b)-f(a)}{b-a} = \frac{f(3)-f(0)}{3-0}}

We can easily see that $f(3) = 1$ and $f(0) = 4$

Thus, the slope is found to be:

$\large{m_{ab} = \frac{1-4}{3-0} = \frac{-3}{3} = -1}$ .

Step 2:

Now, we need to find all the points on the interval which have the same slope. We know that the derivative of the function gives the slope, so, we will find the derivative and set it to -1.

Using the power rule, the derivative of the function is:

\large{f'(x) = 2x-4}

Setting the derivative equal to -1 we have:

\large{2x-4 = -1}

Now, solving for x gives us:

\large{x = \frac{3}{2} }

Thus, $x = \frac{3}{2}$ satisfies the mean value theorem for the given function and interval.

Solution Complete!

Problem: Mean Value Theorem #2

Find all values of $c$ that satisfy the mean value theorem for the function:

$\Large{g(x) = 5x^2+3x-2}$

On the interval $[-1,1]$

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