Question

Verify Rolle's theorem for $f\left(x\right)=x\sqrt{a^2-x^2}$ in $[0,a]$

Answer 1

$f\left(0\right)=0$ $f\left(a\right)=0$ and f is continuous and different able an $[0,a]$ Rolls the can be applied

$\therefore c\in [0,a]$ such that

$f^{\prime }\left(c\right)=0$

$f^{\prime }\left(x_1\right)=\frac{-x^2}{\sqrt{a^2-x^2}}+\sqrt{a^2-x^2}=0$

$\Rightarrow -x^2+a^2-x^2=0$

$x_1=\pm \frac{a}{\sqrt{2}}$

$\therefore c=\frac{a}{\sqrt{2}}$

hence C is found Rollis th verified.