question 5\nfind the value of c for which the function f(x) = sin x on the interval 0, π satisfies the mean…

Explanation:

Step1: Recall Mean - Value Theorem

If $y = f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f^{\prime}(c)=\frac{f(b)-f(a)}{b - a}$ for some $c\in(a,b)$. Here, $a = 0$, $b=\pi$, and $f(x)=\sin x$.

Step2: Calculate $f(a)$ and $f(b)$

$f(0)=\sin(0)=0$, $f(\pi)=\sin(\pi)=0$. So, $\frac{f(\pi)-f(0)}{\pi - 0}=\frac{0 - 0}{\pi}=0$.

Step3: Find the derivative of $f(x)$

The derivative of $f(x)=\sin x$ is $f^{\prime}(x)=\cos x$.

Step4: Solve for $c$

Set $f^{\prime}(c)=\cos c = 0$. Since $c\in(0,\pi)$, then $c=\frac{\pi}{2}$.

Answer:

B. $\frac{\pi}{2}$