Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}={\lim }_{h\to 0}\frac{\frac{f(2x)+f(2h)}{2}-f(x)}{h}$ Putting $x=y=0:f(0)=\frac{2f(0)}{2}$, and putting $y=x:f(x)=\frac{f(2x)+f(0)}{2}\Rightarrow f(2x)=2f(x)-1$ $\Rightarrow f^{\prime }(x)={\lim }_{h\to 0}\frac{f(2h)-1}{2h}=f^{\prime }(0)=-1$ for all $x\Rightarrow f(x)=1-x\Rightarrow f(2)=-1$