Question

(i) Verify the Rolle's theorem for the function
$f\left(x\right)={\sin }^{2}x,0\le x\le \pi$
(ii) Find the critical numbers of $x^{3/5}(4-x)$

Answer 1

(i) $f\left(x\right)={\sin }^{2}x,0\le x\le \pi$
a) $f\left(x\right)$ is continuous on the closed interval $[0,\pi ]$
b) $f\left(x\right)$ is differentiable in the open interval $(0,\pi )$
$f\left(0\right)={\sin }^{2}0$
$f\left(\pi \right)={\sin }^2\pi =0$
$f\left(0\right)=f\left(\pi \right)$
The condition of Rolle's theorem are satisfied
To find $c$:
$f^{\prime }\left(x\right)=2\sin x\cos x$
$f^{\prime }\left(c\right)=0\Rightarrow 2\sin c\cos c=0$
$\sin 2c=0$
$2c=0,\pi ,2\pi ,..$
$c=0,\frac{\pi }{2},\pi ,...$
$c=\frac{\pi }{2}\in (0,\pi )$
The suitable point $c$ is $\frac{\pi }{2}$

(ii) $f\left(x\right)=4x^{3/5}-x^{8/5}$
$f^{\prime }\left(x\right)=\frac{12}{5}{x}^{-2/5}-\frac{8}{5}{x}^{3/5}$
$=\frac{4}{5}{x}^{-2/5}(3-2x)$
$f^{\prime }\left(x\right)=0\Rightarrow 3-2x=0$ $\Rightarrow$ $x=\frac{3}{2}$
$f^{\prime }\left(x\right)$ does not exist when $x=0$
Thus the critical numbers are $0,\frac{3}{2}$.