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GATE 2023 ST – Question 43

Calculus · Single Variable Calculus · 2 marks · Multiple choice

Let g(x)=f(x)+f(2−x) on [0,2],where f is continuous on [0,2] and twice differentiable inside. Which statement is NOT true?

  1. Some c∈(0,2) satisfies g′(c)=0.
  2. If f″>0,then g strictly decreases on (0,1).
  3. If f″<0,then g strictly increases on (1,2).
  4. If f″=0,then g is constant.

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Correct answer: (C) If f″<0,then g strictly increases on (1,2).

Explanation

Differentiate:g′(x)=f′(x)−f′(2−x),so g′(1)=0. For f″>0,f′ increases and x<2−x on (0,1),giving g′<0. For f″<0 and x>1,the decreasing f′ also gives g′<0,so g decreases rather than increases there. Affine f gives constant g. Thus **C** is false.

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