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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Paragraph: Let f: R R, y=f(x), f(0)=0, (x) 0 and f^ (x) 0 . Three points A( , f( )), B( , f( )), C(y, f(y)) on y=f(x) such that 0 Question: Which of the following is false

Options

  1. Af( ) (f( ))
  2. Bf( ) f( )
  3. Cf( ) (f( ))
  4. D(f( )) f( )

Correct answer

B. f( ) f( )

Step-by-step solution

Consider g ( x )= f ( x ) x g^ (x)= f^ (x) x-f(x) x^2 = f^ (x) x^2 (x- f(x) f^ (x) ) as f^ (x) x^2 0 Let h(x)=x- f(x) f^ (x) h^ (x)=1- f^ (x)^2-f(x) f^ (x) f^ (x)^2 = f(x) f^ (x) f^ (x)^2 0 x- f(x) f^ (x) is increasing function aligned & x- f(x) f^ (x) 0- f(0) f^ (0) =0 & g^ (x) 0 & f(y) y f( ) f( ) aligned Satisfies A, C. D

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