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GATE 2022 CE (CE1) – Question 45

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 2 marks · Multiple select

Let $f(x)=\max\{3-x,x-1\}$. Which statements are TRUE?

  1. Continuous on its domain
  2. Local minimum at x=2
  3. Local maximum at x=2
  4. Differentiable on its domain

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Show answer and explanation

Correct answer: (A) Continuous on its domain; (B) Local minimum at x=2

Explanation

The two straight lines intersect where $3-x=x-1$, giving x=2 and f=1. The maximum selects $3-x$ for x<2 and $x-1$ for x>2, forming a V.

It is continuous everywhere and has a minimum at x=2. The one-sided slopes −1 and +1 differ, so it is not differentiable there and has no maximum. **Answer: A and B**.