Let `f:(0, oo) in R and F(x) = int_(0)^(x) f(t) dt`. If `F(x^(2))=x^(2)(1+x)` then f (4) equals

Options

A. `5//4`
B. 7
C. 4
D. 2

Answer

C

Explanation

We have,
`F(x)=underset(0)overset(x) intf(t)dtrArr F'(x) =f(x) rArr f(4)=F'4" "...(i)`
Now,
`F(x^(2))=x^(2)(1+x)=x^(3)+x^(2)`
`rArr F'(x^(2)).2x=3x^(2)+2x` ltbr `rArr F'(x^(2))=(3x+2)/(2)`
`rArr F'(4)=(3xx2+2)/(2)=4 rArr f(4)=4 " " [ "Using (i)"]`
ALTER We have,
`F(x) = underset(0)overset(x)intf(t)dt`
`F(x) = underset(0)overset(x^(2))intf(t)dt rArr x^(2)(1+x)=underset(0)overset(x^(2))int f(t)dt`
Differentiating bhoth sides w.r.t.x. we get
`3x^(2)+2x=f(x^(2)).2x`
`rArr (x^(2)) =(3x+2)/(2) rArr f(4)=4`
ALTER We have,
F(x)`=underset(0)overset(x)int f(t)dt`
`rArr F(x^(2))=underset(0)overset(x^(2))int f (t) dt rArr x^(2)(1+x)=underset(0)overset(x^(2))int f(t) dt`
Differentiating both sides w.r.t.x, we get
`3x^(2)+2x=f(x^(2)).2x`
`rArr F(x^(2))=(3x+2)/(2) rArr f(4)=4`