UPSC NDA I 2025 Mathematics Paper

Concept:A function's increasing or decreasing nature is determined by its derivative. A quadratic with positive leading coefficient has a minimum, not a maximum. Explanation:Given $f(x)=x^{2}+9$. Compute $f'(x)=2x$. For $x>0$, $f'(x)>0$ so $f(x)$ is increasing. For $x<0$, $f'(x)<0$ so $f(x)$ is decreasing. At $x=0$, $f'(x)=0$, which is a stationary point, not a monotonic region. Hence $f(x)$ is not entirely increasing — Statement I is false.Since $f''(x)=2>0$, the function is concave up. So $x=0$ gives a local minimum (global minimum), not a local maximum — Statement II is false.Answer:Neither I nor II (Option D).