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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f: [0, 2 ] [0,1] be a differentiable function such that f(0)=0, f ( 2 )=1 , then

Options

  1. Af^ ( )= 1-(f( ))^2 for all (0, 2 )
  2. Bf^ ( )= 2 for all (0, 2 )
  3. Cf( ) f^ ( )= 1 for at least one (0, 2 )
  4. Df^ ( )= 8 ^2 for at least one (0, 2 )

Correct answer

D. f^ ( )= 8 ^2 for at least one (0, 2 )

Step-by-step solution

(A) Consider g(x)= ⁻¹ f(x)-x since g(0)=0, g ( 2 )=0 there is at least one value of (0, 2 ) such that g ^ ( )= f^ ( ) 1-(f( ))^2 -1=0 i.e. f^ ( )= 1-(f( ))^2 for atleast one value of but may not be for all (0, 2 ) false (B) Consider g(x)=f(x)- 2 x since g(0)=0, g ( 2 )=0 there is at least one value of (0, 2 ) such that g ^ ( )=f^ ( )- 2 =0 i.e. f^ ( )= 2 for atleast one value of but may not be for all aligned & (0, 2 ) & false aligned (C) Consider g ( x )=( f ( x ))^2- 2 x since g(0)=0, g ( 2 )=0 there is at least

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