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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f:(1, ) R be a differentiable function such that ₂¹⁵⁰(x-1) (x-1)(2 f(x)-(x-1) (x-1)) d x= ₂¹⁵⁰ f^2(x) d x . Then:

Options

  1. Aarea bounded by the curve and x -axis is equal to 1 / 2 .
  2. Bf(x) is strictly decreasing in (1,1+ 1 e ) .
  3. Cnumber of solutions of the equation f(x)=2 is 2 .
  4. Df(x) is monotonic in (1+ 1 e , ) .

Correct answer

D. f(x) is monotonic in (1+ 1 e , ) .

Step-by-step solution

Shift LHS integral to RHS to get ₂¹⁵⁰ (f^2(x)-(x-1) (x-1)(2 f(x)-(x-1) (x-1)) ) d x=0 ₂¹⁵⁰ (f^2(x)-2 f(x)(x-1) (x-1)+((x-1) (x-1))^2 ) d x=0 ₂¹⁵⁰(f(x)-(x-1) (x-1))^2 d x=0 array ll But, & (f(x)-(x-1) (x-1))^2 0 So, & (f(x)-(x-1) (x-1))^2=0 & f(x)-(x-1) (x-1)=0 & f(x)=(x-1) (x-1) array (a) is incorrect because area is equal to 1 / 4 .

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