1.

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

Answer»

always equal to 1
always equal to `-2`
may be 1 or `-2`
not independent of x

Solution :`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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