Question

Verify the Rolle's theorem for the function $f\left(x\right)=x^2$ in $(-1,1)$

• A. Yes Rolle's theorem is applicable and the stationary point is $x=\frac{1}{2}$
• B. Yes Rolle's theorem is applicable and the stationary point is $x=0$
• C. No Rolle's theorem is not applicable in the given interval
• D. Both A and B

Answer 1

The correct option is B Yes Rolle's theorem is applicable and the stationary point is $x=0$

$f\left(x\right)=x^2$

$x^2$ is continuous in $[-1,1]$ since it is quadratic equation

$f^{\prime }\left(x\right)=2x$

$2x$ is defined in $(-1,1)$

$\Rightarrow f\left(x\right)$ is differentiable on $(-1,1)$

$f(-1)=f\left(1\right)=1$

There exists a $c,a\le c\le b$ such that

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

$2c=0$

$\Rightarrow c=0$ which lies in $(-1,1)$