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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2018 · Mathematics · Continuity and Differentiability

Consider the function f(x)=\ array cl x^2 (x) & x ≠ 0 0 & x=0 array...

Consider the function \mathrm{f}(\mathrm{x})=\left\{x2ln⁡(x)x≠00x=0\begin{array}{cl}\mathrm{x}^2 \ln (\mathrm{x}) & \mathrm{x} \neq 0 \\ 0 & \mathrm{x}=0\end{array}\right.. What is \mathrmf′(0)\mathrmf^{\prime}(0) equal to?

Options

  1. A0
  2. B1
  3. C–1
  4. DIt does not exist

Correct answer

A. 0

Step-by-step solution

f(x)={x2ℓn∣x∣x≠00x=0f′(0)=lim⁡h→0f(h)−f(0)h=lim⁡h→0h2⋅ln⁡∣h∣−0h=lim⁡h→0h⋅ℓn∣h∣=0\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}
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