Let `f : RR -> RR` be twice continuously differentiable. Let `f(0) = f(1) = f' (0) = 0`.Then

Let f:Rto R be a twice continuously differentiable function such that f(0)=f(1)=f'(0)=0 . Then

Let f RR rarr RR be differentiable at x = 0 . If f(0) = 0 and f'(0) = 2, then the value of underset(x rarr 0)lim (1)/(x)[f(x) + f(2x) + f(3x)+…+f(2015x)] is

Let f is twice differerntiable on R such that f (0)=1, f'(0) =0 and f''(0) =-1, then for a in R, lim _(xtooo) (f((a)/(sqrtx)))^(x)=

Let f is twice differerntiable on R such that f (0)=1, f'(0) =0 and f''(0) =-1, then for a in R, lim _(xtooo) (f((a)/(sqrtx)))^(x)=

Let f:[0,2]to RR be a function which is continuous on [0,2] and is differentiable on (0, 2) with f(0)=1 . Let F(x)=int_(0)^(x^(2))f(sqrt(t))dt" for "x in[0,2] . If F'(x)=f'(x) for all x in(0,2) , the F(2) equals

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then :

Let f defined on [0, 1] be twice differentiable such that | f"(x)| <=1 for x in [0,1] . if f(0)=f(1) then show that |f'(x)<1 for all x in [0,1] .