Question

Verify Rolle's Theorem for the function $f\left(x\right)=x(x-1{)}^2$ in the interval $[0,1]$.

Answer 1

Rolle's Theorem : $\left(i\right)f\left(a\right)=f\left(b\right)$

$\left(ii\right)f\left(x\right)$ is continous in $\left[ab\right]$

$\left(iii\right)f\left(x\right)$ is differential in $(a,b)$

So, $f\left(x\right)=0(0-1{)}^2=0$

$f\left(1\right)=1(1-2{)}^2=0$

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

So , $f\left(x\right)=x(x-1{)}^2$ is satisfy Rolle's theorem.