Q224 / 270 · NDA 2018 (Phase 1) · Mathematics

Consider the function $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cl}\mathrm{x}^2 \ln (\mathrm{x}) & \mathrm{x} \neq 0 \\ 0 & \mathrm{x}=0\end{array}\right.$. What is $\mathrm{f}^{\prime}(0)$ equal to?

  • Correct answer
    A
    0
  • B
    1
  • C
    –1
  • D
    It does not exist

Solution

$\begin{aligned} & f(x)=\left\{\begin{array}{cc}x^2 \ell n|x| & x \neq 0 \\ 0 & x=0\end{array}\right. \\ & f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h} \\ & =\lim _{h \rightarrow 0} \frac{h^2 \cdot \ln |h|-0}{h} \\ & =\lim _{h \rightarrow 0} h \cdot \ell n|h| \\ & =0\end{aligned}$

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

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