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Question: Answered & Verified by Expert
Let $f(x)=\left\{\begin{array}{ll}-2, & -3 \leq x \leq 0 \\ x-2, & 0 < x \leq 3\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$
What is the value of the differential coefficient of $g(x)$ at $x=-2$ ?
MathematicsContinuity and DifferentiabilityNDANDA 2016 (Phase 2)
Options:
  • A -1
  • B 0
  • C 1
  • D 2
Solution:
1639 Upvotes Verified Answer
The correct answer is: 0
For $x=-2$ $g(x)=-2+|-2|=-2+2$
$\Rightarrow g(x)=0$
$\Rightarrow$ differential coefficient at $x=-2$ is given as :
$g^{\prime}(x)=\lim _{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}=\lim _{h \rightarrow 0} \frac{0-0}{4}=0 .$

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