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GATE 2014 CS (CS1) – Question 57

Engineering Mathematics · Calculus · 2 marks · Multiple choice

f is continuous on [0,2], with f(0) = f(2) = −1 and f(1) = 1. Which statement must be true?

  1. Some y in (0,1) satisfies f(y) = f(y+1).
  2. Every y in (0,1) satisfies f(y) = f(2−y).
  3. The maximum of f on (0,2) is 1.
  4. Some y in (0,1) satisfies f(y) = −f(2−y).

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Show answer and explanation

Correct answer: (A) Some y in (0,1) satisfies f(y) = f(y+1).

Explanation

Define g(y)=f(y)−f(y+1) on[0,1]. It is continuous, with g(0)=−2 and g(1)=2. The intermediate value theorem guarantees an interior zero, so f(y)=f(y+1) for some y∈(0,1), **A**. Symmetry and a global maximum of1 are not implied.