Question

Let $f\left(x\right)$ be function satisfying the condition $f(-x)=f\left(x\right)$ for all real.x.If $f^{\prime }\left(0\right)$ exists,find its value.

Answer 1

$f(-x)=f\left(x\right)$ given.
Also $f^{\prime }\left(0\right)$ exists $\therefore Lt\frac{f(0-h)-f\left(0\right)}{h}=Lt\frac{f(0+h)-f\left(0\right)}{h}$
by definition or $\therefore Lt\frac{f\left(h\right)-f\left(0\right)}{-h}=Lt\frac{f\left(h\right)-f\left(0\right)}{h},$
by given condition
$\therefore 2Lt\frac{f\left(h\right)-f\left(0\right)}{-h}=0$ or $2f^{\prime }\left(0\right)=0$
$\therefore f^{\prime }\left(0\right)=0.$
Ans: 0