# Mean Value Theorem (MVT)

Mean Value Theorem hypothesis: Function $$f$$ is continuous on $$[a, b]$$ and differentiable on $$(a, b)$$.

For each function below, circle intervals when the conditions of the MVT are met. For non-circled intervals, provide a reason how the MVT hypothesis fails.

1. $$f(x) = x^2$$, on the intervals

1. $$[4, 9]$$
2. $$[-4, 9]$$
3. $$[0, 9]$$

2. $$f(x) = \sqrt{x}$$, on the intervals

1. $$[4, 9]$$
2. $$[-4, 9]$$
3. $$[0, 9]$$

3. $$f(x) = \frac{1}{x}$$, on the intervals

1. $$[-10, 10]$$
2. $$[1, 10]$$
3. $$[-10, 0]$$

4. $$f(x) = x^{2/3}$$, on the intervals

1. $$[-8, 8]$$
2. $$[0, 8]$$
3. $$[1, 8]$$

Mean Value Theorem result: Then there exists a number $$c \in (a,b)$$ such that

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Choose the first valid interval from the previous page.

1. $$f(x) = x^2$$, on the interval _____
2. $$f(x) = \sqrt{x}$$, on the interval _____
3. $$f(x) = \frac{1}{x}$$, on the interval _____
4. $$f(x) = x^{2/3}$$, on the interval _____

Next, find the number $$c$$ satisfying the conclusion of the MVT for each of the above functions. Hints:

1. Calculate the slope of the secant line between the endpoints, $$m_{sec} = \frac{ f(b)-f(a) }{ b-a }$$
2. Solve for number $$c$$ so that $$f'(c) = m_{sec}$$ where $$c$$ is in the interval $$(a, b)$$

-----

Application: A trucker was given a speeding ticket at a toll booth after handing in her toll booth stub. The stub indicated she had covered 149 miles in 2 hours on a toll road with a speed limit of 65 mph. Was this speeding ticket justified?