lf `f'(x) > 0,f"(x)>0AA x in (0,1)` and `f(0)=0,f(1)=1`,then prove that `f(x)f^-1(x)

If f'(x)>0,f(x)>0AA x in(0,1) and f(0)=0,f(1)=1, then prove that 'f(x)f^(^^)-1(x)

Let f be a one-one function such that f(x).f(y) + 2 = f(x) + f(y) + f(xy), AA x, y in R - {0} and f(0) = 1, f'(1) = 2 . Prove that 3(int f(x) dx) - x(f(x) + 2) is constant.

Let f(x) lt 0 AA x in (-=oo, 0) and f (x) gt 0 AA x in (0,oo) also f (0)=o, Again f'(x) lt 0 AA x in (-oo, -1) and f '(x) gt AA x in (-1,oo) also f '(-1)=0 given lim _(x to oo) f (x)=0 and lim _(x to oo) f (x)=oo and function is twice differentiable. If f'(x) lt 0 AA x in (0,oo)and f'(0)=1 then number of solutions of equatin f (x)=x ^(2) is :

If a function satisfies the relation f(x) f''(x)-f(x)f'(x)=(f'(x))^(2) AA x in R and f(0)=f'(0)=1, then The value of lim_(x to -oo) f(x) is

f(x+f(y))=f(x)+y AA x,y in R and f(0)=1f(x+f(y))=f(x)+y AA x,y in R and f(0)=1 then prove that int_(0)^(2)f(2-x)dx=2int_(0)^(1)f(x)dx

If f(x) is a quadratic expression such that f(x)gt 0 AA x in R , and if g(x)=f(x)+f'(x)+f''(x) , then prove that g(x)gt 0 AA x in R .

Let f(x)=x+f(x-1) for AA x in R.If f(0)=1, find f(100)

If f(x) is continuous in [0,2] and f(0)=f(2). Then the equation f(x)=f(x+1) has