Let `f : R rarr R` be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of `underset(x rarr 0)(lim) (1)/(x) . underset(n = 1)overset(oo)(sum)(-1)^(n).f((x)/(n))`, is

Let f : R rarr R be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of lim_(x to 0) (1)/(x) . sum_(n=1)^(oo)(-1)^(n).f((x)/(n)) , is

Let f : R to R be a differentiable function satisfying f'(3) + f'(2) = 0 , Then underset(x to 0) lim ((1+f(3+x)-f(3))/(1+f(2-x)-f(2)))^(1/x) is equal to

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then f(x) is

Let f(x) be a differentiable function satisfying f(y)f(x/y)=f(x) AA , x,y in R, y!=0 and f(1)!=0 , f'(1)=3 then

Let f be a continuous function satisfying f '(l n x)=[1 for 0 1 and f (0) = 0 then f(x) can be defined as

Let f: R rarr R be a differentiable function satisfying f(x+y)=f(x)+f(y)+x^(2)y+xy^(2) for all real numbers x and y. If lim_(xrarr0) (f(x))/(x)=1, then The value of f(9) is

If f:R->R is a twice differentiable function such that f''(x) > 0 for all x in R, and f(1/2)=1/2 and f(1)=1, then

If a continuous function f on [0,a] satisfies f(x)f(a-x)=1,agt0 , then find the value of int_(0)^(a)(dx)/(1+f(x)) .

Let f:(0,1) in (0,1) be a differenttiable function such that f(x)ne 0 for all x in (0,1) and f((1)/(2))=(sqrt(3))/(2) . Suppose for all x, underset(x to x)lim(overset(1)underset(0)int sqrt(1(f(s))^(2))dxoverset(x)underset(0)int sqrt(1(f(s))^(2))ds)/(f(t)-f(x))=f(x) Then, the value of f((1)/(4)) belongs to