UPSC NDA II 2025 Mathematics Paper

Concept:The given function is defined as $f(x)=x^{2}$ for $x\ge0$ and $f(x)=-x^{2}$ for $x<0$. Its graph shows it is always increasing and smooth at $x=0$.Explanation:From the graph, $f(x)$ rises from left to right for all $x$, so it is increasing in $(-\infty,\infty)$. Left-hand derivative at $x=0$: $\lim_{h\to0^-}\frac{f(0+h)-f(0)}{h} = \lim_{h\to0^-}\frac{-h^{2}}{h}=0$.Right-hand derivative at $x=0$: $\lim_{h\to0^+}\frac{h^{2}}{h}=0$.Since both derivatives are equal ($0$), the function is differentiable at $x=0$.Answer:Both statements I and II are correct. Hence, option C (Both I and II).