Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

If f(x)= ₀^x[f(x)]⁻¹ d x and ₁^1 [ .f(x) |⁻¹ d x= 2 . , then

Options

Af(2)=2
Bf(2)=1 / 2
Cf^ (2)=2
D₀^1 f(x) d x= 2

Correct answer

C. f^ (2)=2

Step-by-step solution

f(x)= _a^x 1 f(x) d x f(x)= 1 f(x) 1-0 f(x) f(x)=1 f ( x ) f ^ ( x ) dx = 1 dx ( f ( x ))^2 2 =x+c ....(1) Given _a^1[f(x)]⁻¹ d x= 2 f(1)= 2 From (1), (f(1))^2 2 =1+c c=0f(x)= 2 x But f(1)= 2 f(x)= 2 x f(2)=2f^ (x)= 1 2 x f^ (2)= 1 2 ₀^1 f(x) d x= 2^ 3 / 2 3 f⁻¹(x)= x^2 2 f⁻¹(2)=2

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