Answer
(b) $\sqrt{3}$
(a) $ \pi, \pm 2\pi, \pm 3\pi, \pm 4\pi, ... $
Work Step by Step
Mean-Value Theorem for Integrals: $\int_{a}^{b} f(x) d x=f\left(x^{*}\right)(-a+b)$
(a) $\int_{-\pi}^{\pi} \sin x d x=0$
$\Rightarrow 0= f\left(x^{*}\right).2 \pi$
$\Rightarrow 0=\sin x^{*}=f\left(x^{*}\right)$
$\Rightarrow \pm \pi, \pm 2\pi, \pm 3\pi, \pm 4\pi, ... =x^{*}$
(b) $\int_{1}^{3} \frac{1}{x^{2}} d x=\frac{2}{3}$
$\Rightarrow \frac{2}{3}= f\left(x^{*}\right).2$
$\Rightarrow \frac{1}{3}=\frac{1}{\left(x^{*}\right)^{2}}=f\left(x^{*}\right)$
$\Rightarrow \sqrt{3}=x^{*}$