Question

Verify Rolle's theorem for the function $y=x^2+2$. $xϵ|-2,2|$.

Answer 1

Given that $y=x^2+2$ and $x$ belongs to $[-2,2]$

Here $f\left(x\right)$ is continuous on a closed interval $[-2,2]$ and differentiable on an open interval $(-2,2)$

We have $f\left(2\right)=f(-2)=6$

According to rolles theorem , if $f(-2)=f\left(2\right)$ then there exists at least one point $c$ in $(-2,2)$ such that $f^{\prime }\left(c\right)=0$

Now, to check whether such $c$ exists or not

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

$f^{\prime }\left(x\right)=2x=0$ for $x=0$, and $-2<0<2$

Hence, there exist $0\in (-2,2)$ such that $f^{\prime }\left(0\right)=0$

Therefore, Rolle's theorem is verified.