Let f be a real-valued function of a real variable defined as f(x) = x^{2 }for x ≥ 0, and f(x) = -x^{2} for x < 0.

Which one of the following statements is true?

This question was previously asked in

GATE EE 2018 Official Paper

Option 4 : f(x) is differentiable but its first derivative is not differentiable at x = 0.

ST 2: Strength of materials

2026

15 Questions
15 Marks
15 Mins

Given that,

A function f(x) is said to be differentiable at x =a if,

Left derivative = Right derivative = Well defined

i.e.,

\(\rm\lim _{x \rightarrow a^{-}} f'(x)=\lim _{x \rightarrow a^{+}} f'(x)\)

**Analysis:**

f(x) = x^{2}, x ≥ 0

= -x^{2}, x ≤ 0

f^{'}(x) = 2x, x ≥ 0

= -2x, x < 0

f^{'}(x) = 2|x|

f^{’}(x) is continuous but not differentiable at x = 0.

Hence f(x) is differentiable but its first derivative is not differentiable at x = 0.