Question

For $x\in [-1,1],$ Rolle's theorem can be applicable on

• A. $\text{sgn}\left(x\right)$
• B. $x|x|$
• C. $x^2$
• D. $x^3$

Answer 1

The correct option is C $x^2$

For Rolle's theorem to be applicable function should be continuous in $[-1,1]$ and differentiable in $(-1,1)$ and $f(-1)=f\left(1\right)$

Now, $f_1\left(x\right)=\text{sgn}\left(x\right)$ is discontinuous at $x=0$

$f_2\left(x\right)=x|x|$
$f_2\left(x\right)$ is continuous and differentiable for $x\in R$
$f\left(1\right)=1\ne f(-1)=-1$

$f_3\left(x\right)=x^2,$ is continuous and differentiable everywhere also $f(-1)=f\left(1\right)$

$f_4\left(x\right)=x^3$
$f_4\left(x\right)$ is continuous and differentiable for $x\in R$
$f(-1)\ne f\left(1\right)$

$\therefore$ On $x^2,$ Rolle's theorem is applicable for $x\in [-1,1]$