Consider the following statements for the curve $f(x) = x|x|$ and find that which of the following(s) are/is correct :

(A) $f(x) = x|x|$ is differentiable at $x = 0$.

(B) $f(x) = x|x|$ is continuous at $x = 0$ but not differentiable at $x = 0$.

(C) $f(x) = x|x|$ has point of infection at $x = 0$.

(D) $f(x) = x|x|$ is symmetrical about y-axis.

Choose the correct answer from the options given below :

$f(x) = x^2$ for $x \geq 0$, $-x^2$ for $x < 0$. $f'(x) = 2x$ for $x > 0$, $-2x$ for $x < 0$. Both give $0$ at $x = 0$, so (A) differentiable: TRUE.

(B) False since $f$ IS differentiable at $0$.

(C) $f''$ changes sign at $0$ (from $-2$ to $2$), so point of inflection: TRUE.

(D) $f(-x) = -f(x)$ (odd function), symmetric about origin not y-axis: FALSE.

So (A) and (C) only.

Note: 'point of infection' in the question is interpreted as 'point of inflection'.