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GATE 2024 EE – Question 44

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 2 marks · Multiple select

Consider the function $f(t)=(\max(0,t))^2$ for $-\infty<t<\infty$, where $\max(a,b)$ denotes the maximum of $a$ and $b$. Which of the following statements is/are true?

  1. $f(t)$ is not differentiable.
  2. $f(t)$ is differentiable and its derivative is continuous.
  3. $f(t)$ is differentiable but its derivative is not continuous.
  4. $f(t)$ and its derivative are differentiable.

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Correct answer: (B) $f(t)$ is differentiable and its derivative is continuous.

Explanation

$f(t)=t^2$ for $t\ge0$ and 0 for $t<0$. The derivative is $2t$ for $t>0$ and 0 for $t<0$, and at $t=0$ both one-sided derivatives equal 0, so $f'(t)=2\max(0,t)$ exists everywhere and is continuous. But $f'$ has slope 2 on the right and 0 on the left of the origin, so it is not differentiable at $t=0$ (D is false).