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JEE Main20209 Jan 2020Morning ShiftMathematicsDifferentiationActual

Let f be any function continuous on a , b and twice differentiable on a , b . If all x ∈ a , b , f ' x > 0 and f ' ' x < 0 , then for any c ∈ a , b , f c - f a f b - f c

Options

  1. Ab + a b - a
  2. B1
  3. Cb - c c - a
  4. Dc - a b - c

Correct answer

D. c - a b - c

Step-by-step solution

Let’s use LMVT for x ∈ a , c f c - f ( a ) c - a = f ' α , α ∈ a , c Also use LMVT for x ∈ c , b f b - f ( c ) b - c = f ' β , β ∈ c , b ∵ f ' ' x < 0 ⇒ f ' x is decreasing f ' α > f ' β f c - f ( a ) c - a > f b - f ( c ) b - c f c - f ( a ) f b - f ( c ) > c - a b - c ( ∵ f ( x ) is increasing)

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