The Mean Value Theorem — Question 10

Question 10

Problem:

Let $$f(x) = \sin(x)$$ on the interval $\left[0, \pi\right]$.

• Verify that $f(x)$ satisfies the conditions of the Mean Value Theorem on $\left[0, \pi\right]$.

• Find all values of $c \in (0, \pi)$ such that $$f'(c) = \frac{f(\pi) - f(0)}{\pi - 0}.$$

Question 10 - Solution

We are given: $$f(x) = \sin(x)$$

(a) Conditions for MVT:

- $f(x)$ is continuous on $[0, \pi]$ ✓ - $f(x)$ is differentiable on $(0, \pi)$ ✓

✅ MVT applies.

(b) Apply MVT:

$$f(\pi) = 0, \quad f(0) = 0$$ $$\frac{f(\pi) - f(0)}{\pi - 0} = \frac{0 - 0}{\pi} = 0$$

We need: $$f'(c) = \cos(c) = 0 \Rightarrow c = \frac{\pi}{2}$$

$$\boxed{c = \frac{\pi}{2}}$$

Graph of $f(x) = \sin(x)$ on $[0, \pi]$: