Question

Is Rolle's theorem valid for the function $y=x^3+4x^2-7x-10$ in the interval $[-1,2].$

Answer 1

Given $y=f\left(x\right)=x^3+4x^2-7x-10$

Since, $f\left(x\right)$ is a polynomial function. So, it is continuous everywhere.

$\Rightarrow f\left(x\right)$ is continuous on [-1,2]

Also, $f^{\prime }\left(x\right)=3x^2+8x-7$

$f^{\prime }\left(x\right)$ exists uniquely on (-1,2)

$f(-1)=-1+4+7-10=0$

$f\left(2\right)=8+16-14-10=0$

So, all the three conditions of Rolle's theorem holds.