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) lt x^2AA x in (0,1)`

If f(x+f(y))=f(x)+yAAx ,y in R and f(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(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 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.

If f'(x) = f(x)+ int _(0)^(1)f (x) dx and given f (0) =1, then int f (x) dx is equal to :

Let f(x) lt 0 AA x in (-oo, 0) and f (x) gt 0 ,AA x in (0,oo) also f (0)=0, Again f'(x) lt 0 ,AA x in (-oo, -1) and f '(x) gt 0, 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 equation f (x)=x ^(2) is : (a) 1 (b) 2 (c) 3 (d) 4

Let f(x) lt 0 AA x in (-oo, 0) and f (x) gt 0 AA x in (0,oo) also f (0)=0, Again f'(x) lt 0 AA x in (-oo, -1) and f '(x) gt 0 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. The minimum number of points where f'(x) is zero is: (a) 1 (b) 2 (c) 3 (d) 4

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] .

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

If f(x) is positive for all positive values of X and f'(x)lt0,f"(x)gt0,AA x in R^(+) , prove that 1/2f(1)+int_(1)^(n)f(x)dx lt sum_(r=1)^(n)f(r) lt int_(1)^(n) f(x)dx+f(1) .

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 Number of roots of the equation f(x)=e^(x) is