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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 |
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Answer» always equal to 1 `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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