If `a , b` are two fixed positive integers such that `f(a+x)=b+[b^3+1-3b^2f(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.

If f(x) satisfies the relation f(x)+f(x+4)=f(x+2)+f(x+6) for all x , then prove that f(x) is periodic and find its period.

If f: Rrarr[0,oo) is a function such that f(x-1)+f(x+1)=sqrt(3)f(x), then prove that f(x) is periodic and find its period.

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.

Given that f'(x)=4x^(3)-3x^(2)+2x-1 find f(x) if f(0)=0 .

H-7.Let'f' be a real valued function defined for all real numbers x such that for some positive constant 'a' theequation f(x+a)= 2 1 ​ + f(x)−(f(x)) 2 holds for all x. Prove that the function f is periodic.

If f(x+1/2)+f(x-1/2)=f(x) for all x in R , then the period of f(x) is 1 (b) 2 (c) 3 (d) 4

If f((x+y)/3)=(2+f(x)+f(y))/3 for all x,y f'(2)=2 then find f(x)

If f is a function such that f(0)=2,f(1)=3,a n df(x+2)=2f(x)-f(x+1) for every real x , then f(5) is 7 (b) 13 (c) 1 (d) 5

If f(x)=x^3+b x^2+c x+da n df(0),f(-1) are odd integers, prove that f(x)=0 cannot have all integral roots.

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot