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GATE 2025 EE – Question 11

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

Consider the set $S$ of points $(x,y)\in\mathbb R^2$ which minimize the real valued function

$$f(x,y)=(x+y-1)^2+(x+y)^2$$

Which of the following statements is true about the set $S$?

  1. The number of elements in the set $S$ is finite and more than one.
  2. The number of elements in the set $S$ is infinite.
  3. The set $S$ is empty.
  4. The number of elements in the set $S$ is exactly one.

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Show answer and explanation

Correct answer: (B) The number of elements in the set $S$ is infinite.

Explanation

$f$ depends only on $s=x+y$: $f=(s-1)^2+s^2$, which is minimised at $s=\tfrac12$. Every point on the line $x+y=\tfrac12$ is a minimiser, so $S$ is infinite.