Question

$f\left(x\right)=x^4-3x^2+4$ in the interval [-4, 4]. Is Rolle's theorem applicable?

Answer 1

$f\left(x\right)=x^4-3x^2+4$ in interval [-4, 4]
Since, $f\left(x\right)$ is a polynomial, therefore it is continuous as well as differentiable in the given interval.

And $f^{\prime }\left(x\right)=4x^3-6x=0$

$\Rightarrow x=0,\pm \sqrt{\frac{3}{2}}ϵ(-4,4)$

Also $f(-4)=f\left(4\right).$

Therefore, all conditions of Rolle's theorem are satisfied.
Hence, Rolle's theorem is applicable.