Question

Verify Rolle's theorem for the following function $f\left(x\right)=x^2(1-x^2)$ on $[0,1]$, if it is applicable, then find $c$.

Answer 1

The given function $f\left(x\right)=x^2(1-x^2)$ on $[0,1]$ is continuous and differentiable on $(0,1)$, since it is a polynomial of degree $4$.

Again $f\left(0\right)=0=f\left(1\right)$.

So $f\left(x\right)$ satisfies all the conditions of Rolle's theorem.

According to Rolle's theorem there exists $c\in (0,1)$ such that $f^{\prime }\left(c\right)=0$.

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

$2c-4c^3=0$

or, $2c(1-2c^2)=0$

or, $c=\frac{1}{\sqrt{2}}$ [ Since $c\in (0,1)$]