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Let f(x) be a differentiable non-decreas...

Let f(x) be a differentiable non-decreasing function such that `int_(0)^(x)(f(t))^(3)dt=(1)/(x^(2))(int_(0)^(x)f(x)dt)^(3)AAx inR-{0} andf(1)="1. If "int_(0)^(x)f(t)dt=g(x)" then "(xg'(x))/(g(x))` is

A

always equal to 1

B

always equal to `-2`

C

may be 1 or `-2`

D

not independent of x

Text Solution

Verified by Experts

The correct Answer is:
A
`int_(0)^(x)(f(t))^(3)dt=(1)/(x^(2))(int_(0)^(x)f(t)dt)^(3)`
`therefore" "int_(0)^(x)(f(t))^(3)dt=(1)/(x^(3))(g(x))^(3)`
Differentiating w.r.t. x,
`therefore" "(f(x))^(3)=(1)/(x^(2))3(g(x))^(2)g'(x)-(2)/(x^(3))(g(x))^(3)`
`rArr" "((xg'(x))/(g(x)))^(3)-3((x.g'(x))/(g(x)))+2=0`
`rArr" "(xg'(x))/(g(x))=1or -2`
If `(xg'(x))/(g(x))=1`
`rArr" "xf(x)=int_(0)^(x)f(t)dt`
`rArr" "xf'(x)+f(x)=f(x)`
`rArr" "f(x)=1`
`"or "xf'(x)+f(x)=-2f(x)`
`" "(f'(x))/(f(x))=(-3)/(x)`
`logf(x)=-3log x+logc`
`rArr" "f(x)=c//x^(3)`
`rArr" "f(1)=1 rArr f(x)=1//x^(3)" (decreasing function)"`
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