Most Important Selected Qs for JEE AdvancedMathematicsFunctions

Let y=f(x) be function defined as x=y^3+y^2+y+1 , then which of the following is/are correct?

Options

A2 f^ (0)=1
Bf^ (0)= 1 2
C₀^4 f(x) d x= 4 3
D₀^4 f(x) d x=0

Correct answer

C. ₀^4 f(x) d x= 4 3

Step-by-step solution

Let g(x)=f⁻¹(x) g(x)=x^3+x^2+x+1f^ (x)= 1 g^ (f(x)) and f^ (x)= -g^ (f(x)) (g^ (f(x)) )^3 array ll & f^ (0)= 1 g^ (f(0)) = 1 g^ (-1) = 1 2 & f^ (0)= -g^ (-1) ( 1 2 )^3 =32 array I= ₀^4 f(x) d x put x=g(t)I= _ -1 ^1 t (3 t^2+2 t+1 ) d t= 4 3

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