Does there exist a differentiable function f(x) such that `f(0) = -1, f(2) = 4 and f'(x) le 2` for all x. Justify your answer.

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x)le2 for all x. Justify your answer.

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

Let f(x) and g(x) are two differentiable functions. If f(x)=g(x) , then show that f'(x)=g'(x) . Is the converse true ? Justify your answer

Let f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)), then h'(1) is equal to

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 : R to R be a thrice differentiable function . Suppose that F(1)=0,F(3)=-4 and F(x) lt 0 " for all" x in (1/2,3). f(x) = x F(x) for all x inR . The correct statement (s) is / are

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

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. f(1)=1, then

Let f be a differentiable function from R to R such that abs(f(x)-f(y))abs(le2)abs(x-y)^(3//2) ,for all x,y inR .If f(0)=1 ,then int_(0)^(1)f^2(x)dx is equal to

Let f be a differential function such that f(x)=f(2-x) and g(x)=f(1 +x) then (1) g(x) is an odd function (2) g(x) is an even function (3) graph of f(x) is symmetrical about the line x= 1 (4) f'(1)=0