Let `f(x)` be differentiable on the interval `(0,oo)` such that `f(1)=1` and `lim_(t->x) (t^2f(x)-x^2f(t))/(t-x)=1` for each `x>0`. Then `f(x)=`

Let f(x) be differentiable on the interval (0,oo) such that f(1)=1 and lim_(t rarr x)(t^(2)f(x)-x^(2)f(t))/(t-x)=1 for each x>0 Then f(x)=

Let f:(0, oo)rarr(0, oo) be a differentiable function such that f(1)=e and lim_(trarrx)(t^(2)f^(2)(x)-x^(2)f^(2)(t))/(t-x)=0 . If f(x)=1 , then x is equal to :

Let f (x) be a diffentiable function in [-1,oo) and f (0) =1 such that Lim _(t to x +1) (t^(2) f(x+1) -(x+1) ^(2) (t))/(f (t) -f(x+1))=1. Find teh value of Lim _(x to 1) (ln (f(x )) -ln 2)/(x-1)

Let f(x) be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt then int_(0)^(1)f(x)dx=

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x) and f'((1)/(4))=0. Then (a) f'(x) vanishes at least twice on [0,1](b)f'((1)/(2))=0( c) int_(-(1)/(2))^((1)/(2))f(x+(1)/(2))sin xdx=0 (d) int_(-(1)/(2))^((1)/(2))f(t)e^(sin pi t)dt=int_((1)/(2))^(1)f(1-t)e^(sin pi t)dt

if f be a differentiable function such that f(x) =x^(2)int_(0)^(x)e^(-t)f(x-t). dt. Then f(x) =