UPSC NDA I 2024 Mathematics Paper

Concept:For a function $f(x)$ to have a local maximum at $x = 0$, the first derivative must be zero and the second derivative must be negative (sufficient condition).Explanation:Given that $f(x)$ has a local maximum at $x = 0$.By definition of a local maximum, the slope of the tangent at that point is zero.Thus, $f'(0) = 0$. So statement 1 is correct.For a local maximum, the function should be concave downward at that point.This requires the second derivative to be negative, i.e., $f''(0) < 0$. So statement 2 is correct.Both conditions together confirm a local maximum at $x = 0$.Answer:Both statements 1 and 2 are correct. Hence option C.