Which of the following functions \(f: \mathbb{R} \to \mathbb{R}\) has/have a local minimum at \(x = 0\)?
IIT JAM - 2026,
Answer: A, C, D
A local minimum at \(x = 0\) occurs in (A), (C) and (D).
(A) \(f(x) = \sin|x|\) satisfies \(f(0) = 0\) and \(\sin|x| \gt 0\) for small \(x \ne 0\), so 0 is a local minimum. (C) \(x^{4} + x^{2} + 3 \ge 3 = f(0)\) for all \(x\). (D) For \(0 \lt x \lt 1\), \(f(x) = \min\{x, 1 - x\} \gt 0\); for \(-1 \lt x \lt 0\) we have \([x] = -1\), so \(f(x) = \min\{x + 1, -x\} = -x \gt 0\); and \(f(0) = 0\). So \(f \ge 0 = f(0)\) near 0.
(B) \(f'(x) = \cos x + \frac{x^{2}}{2} \gt 0\) near 0, because \(\cos x\) is close to 1. The function is increasing there, so \(x = 0\) is not an extremum.
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