GATE 2020 AE – Question 11

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 1 mark · Multiple choice

For $f(x)=|x|$, with df/dx denoting the derivative, the mean value theorem is not applicable because

f(x) is not continuous at x = 0.
f(x) = 0 at x = 0.
df/dx is not defined at x = 0.
df/dx = 0 at x = 0.

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Correct answer: (C) df/dx is not defined at x = 0.

Explanation

On an interval crossing zero, |x| is continuous but has left derivative −1 and right derivative +1 at zero. It is therefore not differentiable at the interior point x = 0, violating the differentiability hypothesis of the mean value theorem. Select C. On an interval entirely on one side of zero the theorem would apply.

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