Question

(i) Verify Rolle's theorem for the function
$f\left(x\right)=|x|,-1\le x\le 1$.
(ii) Obtain the Maclaurin's series for the function $e^x$.

Answer 1

(i) $f\left(x\right)=|x|-1\le x\le 1$
$f$ is continuous in $[-1,1]$, but not differentiable in $(-1,1)$, since $f^{\prime }\left(0\right)$ does not exist
Thus Rolles thorem is not applicable.
(ii) $f\left(x\right)=e^x;f\left(0\right)=e^0=1$
$f^{\prime }\left(x\right)=e^x;f^{\prime }\left(0\right)=1$
$f"\left(x\right)=e^x;f"\left(0\right)=1$
$f\left(x\right)=e^x=1+\frac{1x}{1!}+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+........$
$=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+.......$ holds for all $x$