If `f(a-x)=f(a+x) " and " f(b-x)=f(b+x)` for all real x, where `a, b (a gt b gt 0)` are constants, then prove that `f(x)` is a periodic function.

Let f(x) =ax^(2) -b|x| , where a and b are constants. Then at x = 0, f (x) is

If f(x+y)=f(x)*f(y) for all real x,y and f(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^(2)) is an even function.

Let f(x)=ax^(2)-b|x| , where a and b are constant . Then at x=0 , f(x) has

Let f(x+y)=f(x)+f(y)+2xy-1 for all real x and f(x) be a differentiable function.If f'(0)=cosalpha, the prove that f(x)>0AA x in R

If f(a+b+1-x)=f(x) , for all x where a and b are fixed positive real numbers, the (1)/(a+b) int_(a)^(b) x(f(x)+f(x+1) dx is equal to :

If a,b are two fixed positive integers such that f(a+x)=b+[b^(3)+1-3b^(2)f(x)+3b{f(x)}^(2)-{f(x)}^(3)]^((1)/(3)) for all real x, then prove that f(x) is periodic and find its period.

A function is defined as f(x) = ax^(2) – b|x| where a and b are constants then at x = 0 we will have a maxima of f(x) if

Let f(x)=a+b|x|+c|x|^(2) , where a,b,c are real constants. The, f'(0) exists if

Let f be a function defined on [a,b] such that f'(x)>0 for all quad x in(a,b) ( Then prove that f is an increasing function on (a,b).