The GATE Grind

GATE 2024 DA – Question 15

Calculus and Optimization · Taylor series, maxima and minima, single-variable optimization · 1 mark · Multiple choice

For any twice differentiable function $f: \mathbb{R} \to \mathbb{R}$, if at some $x^* \in \mathbb{R}$, $f'(x^*) = 0$ and $f''(x^*) > 0$, then the function $f$ necessarily has a ______ at $x = x^*$.

  1. local minimum
  2. global minimum
  3. local maximum
  4. global maximum

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Show answer and explanation

Correct answer: (A) local minimum

Explanation

By the second derivative test, a stationary point where $f'' > 0$ is a local minimum. Nothing says that it is the lowest value over the whole real line, so it need not be a global minimum.