Q254 / 270 · NDA 2013 (Phase 2) · Mathematics

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $f(x)=x^2$ for $x \geq$ 0 and $f(x)=-x$ for $x \lt 0$. Consider the following statements in respect of the above function: 1. The function is continuous at $x=0$. 2. The function is differentiable at $x=0$. Which of the above statements is/are correct?

  • Correct answer
    A
    1 only
  • B
    2 only
  • C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

Test of continuity at $x=0$ $\mathrm{R} f(x)=\lim _{x \rightarrow 0^{+}} x^2=0$ $\mathrm{L} f(x)=\lim _{x \rightarrow 0^{-}}(-x)=0$ also, $f(0)=(0)^2=0$ Hence, $f(x)$ is continuous at $x=0$ Differentiability $f^{\prime}(x)$ for $(x \gt 0)=2 x \Rightarrow f^{\prime}(0)=0$ $f^{\prime}(x)$ for $(x \lt 0)=-1 \Rightarrow f^{\prime}(0)=-1$ Hence, $f(x)$ is not differentiable at $x=0$.

Hence, the correct answer is option: 'A'.

Relevant Questions