$f'(x)=\lim\limits_{h\to 0}\large\frac{f(x+h)-f(x)}{h}$
$\mid f'(x)\mid=\lim\limits_{h\to 0}\mid\large\frac{f(x+h)-f(x)}{h}\mid$$\leq \lim\limits_{h\to 0}\mid \large\frac{(h)^2}{h}\mid$
$\Rightarrow \mid f'(x)\mid\leq 0\Rightarrow f'(x)=0$
$\Rightarrow f(x)$=constant
As $f(0)=0$
$\Rightarrow f(1)=0$
Hence (b) is the correct answer.