Question

Rolle's theorem is applicable in the interval $[-2,2]$ for the function

• A. $f\left(x\right)=x^3$
• B. $f\left(x\right)=4x^4$
• C. $f\left(x\right)=2x^3+3$
• D. $f\left(x\right)=\pi |x|$

Answer 1

The correct option is B $f\left(x\right)=4x^4$

If we take $f\left(x\right)=4x^4$, then
(i) $f\left(x\right)$ is continuous in $(-2,2)$
(ii) $f\left(x\right)$ is differentiable in $(-2,2)$
(iii) $f(-2)=f\left(2\right)$
So, $f\left(x\right)=4x^4$ satisfies all the conditions of Rolle's theorem, therefore a point $c$ such that $f^{\prime }\left(c\right)=0$
$\Rightarrow 16c^3=0\Rightarrow c=0ϵ(-2,2)$