Which of the following functions has a local minima at \(x = 0\)?
(A) \(f(x)=x^3\)
(B) \(f(x)=|x|\)
(C) \(f(x)= x^2\)
(D) \(f(x)=x^{-2}\)
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
f has a local minimum at x = 0 when f is defined at 0 and
\(f(x) \ge f(0)\) for all x near 0.
So x = 0 is a local minimum of |x| and \(x^2\) (both are 0 there and positive elsewhere), but not of \(x^3\) (negative for x < 0) or of \(x^{-2}\) (not defined at 0).
Checking the statements
(A) \(f(x)=x^3\) — Incorrect. \(x^3 \lt 0 = f(0)\) for x < 0; x = 0 is a point of inflexion.
(B) \(f(x)=|x|\) — Correct. \(|x| \ge 0 = f(0)\) for all x, though f is not differentiable at 0.
(C) \(f(x)= x^2\) — Correct. \(x^2 \ge 0 = f(0)\); also \(f'(0) = 0\), \(f''(0) = 2 \gt 0\).
(D) \(f(x)=x^{-2}\) — Incorrect. \(x^{-2} = \frac{1}{x^2}\) is not defined at x = 0.