Mathematics - Application of Derivative Question with Solution | TestHub

$$\begin{array}{rl}& f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\\ & \left|f^{\prime }(x)\right|=\underset{h\to 0}{\lim }\left|\frac{f(x+h)-f(x)}{h}\right|\le \underset{h\to 0}{\lim }\left|\frac{(h{)}^2}{h}\right|\\ & \Rightarrow \left|f^{\prime }(x)\right|\le 0\Rightarrow f^{\prime }(x)=0\Rightarrow f(x)=\text{ constant }\\ & \text{ As }f(0)=0\Rightarrow f(1)=0.\end{array}$$