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.

(B) and (C) only

The correct answer is Option (4) → (B) and (C) only

(A) $f(x)=x^{3}$ → $f'(x)=3x^{2}$, $f''(x)=6x$, $f''(0)=0$ → point of inflection, no minimum.

(B) $f(x)=|x|$ → decreases for $x<0$, increases for $x>0$ → local minimum at $x=0$.

(C) $f(x)=x^{2}$ → $f'(x)=2x$, $f'(0)=0$, $f''(0)=2>0$ → local minimum at $x=0$.

(D) $f(x)=x-2$ → linear function, no extremum.

Functions having local minimum at $x=0$ are (B) and (C).