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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f(x) be a polynomial function satisfying 0 < x f(y) < y f(x) x, y such that 0 < x < y < 1 and f(0)=0 then:

Options

  1. Af^ (x) < f(1)
  2. Bf(1) < 2 ₀^1 f(x) d x
  3. C3 f ( 1 3 )>2 f ( 1 2 )
  4. D6 f ( 1 6 ) < 5 f ( 1 5 )

Correct answer

C. 3 f ( 1 3 )>2 f ( 1 2 )

Step-by-step solution

x f(y) < y f(x) 0 < x < y < 1 aligned & f(y) y < f(x) x x < y & g(x)= f(x) x is decreasing & g^ (x) < 0 x f^ (x) < f(x) & f^ (x) < f(x) x x 0 < x < 1 aligned Hence, f^ (x) < g(x) x 0 < x < 1 Hence, f^ (x) is less than minimum value of g(x) which is g(1) . f^ (x) < f(1) ...(1) x f^ (x) < f(x) ₀^1 x f^ (x) d x < ₀^1 f(x) d x . x f(x) |₀ ^1- ₀^1 f(x) d x < ₀^1 f(x) d x f(1) < 2 ₀^1 f(x) d x ...(2)

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