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GATE 2018 EE – Question 21

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 1 mark · Multiple choice

Let $f$ be a real-valued function of a real variable defined as $f(x)=x^2$ for $x\ge0$, and $f(x)=-x^2$ for $x<0$. Which one of the following statements is true?

  1. $f(x)$ is discontinuous at $x=0$.
  2. $f(x)$ is continuous but not differentiable at $x=0$.
  3. $f(x)$ is differentiable but its first derivative is not continuous at $x=0$.
  4. $f(x)$ is differentiable but its first derivative is not differentiable at $x=0$.

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Correct answer: (D) $f(x)$ is differentiable but its first derivative is not differentiable at $x=0$.

Explanation

$f'(x)=2|x|$, which is continuous and equals 0 at $x=0$, so $f$ is differentiable there. But $2|x|$ has a corner at $0$, so $f'$ is not differentiable.