The GATE Grind

GATE 2021 EE – Question 13

Engineering Mathematics · Calculus: Maxima and minima · 1 mark · Multiple choice

Let $f(x)$ be a real-valued function such that $f'(x_0)=0$ for some $x_0\in(0,1)$, and $f''(x)>0$ for all $x\in(0,1)$. Then $f(x)$ has

  1. no local minimum in $(0,1)$
  2. one local maximum in $(0,1)$
  3. exactly one local minimum in $(0,1)$
  4. two distinct minima in $(0,1)$

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Correct answer: (C) exactly one local minimum in $(0,1)$

Explanation

Since $f''>0$ throughout, $f$ is strictly convex, so $f'$ is strictly increasing and vanishes only at $x_0$. There $f''>0$, so $x_0$ is the only stationary point and it is a local minimum.