A function `f:R->R` satisfies the relation `f((x+y)/3)=1/3|f(x)+f(y)+f(0)|` for all `x,y in R.` If `f'(0)` exists, prove that `f'(x)` exists for all `x, in R.`

A function f(x) satisfies the relation f(x+y) = f(x) + f(y) + xy(x+y), AA x, y in R . If f'(0) = - 1, then

If f(x) satisfies the relation, f(x+y)=f(x)+f(y) for all x,y in R and f(1)=5, then find sum_(n=1)^(m)f(n) . Also, prove that f(x) is odd.

A function f: R rarr R satisfies the equation f(x+ y)= f(x).f(y) "for all" x, y in R, f(x) ne 0 . Suppose that the function is differentiable at x=0 and f'(0)=2, then prove that f'(x)= 2f(x)

Let f((x+y)/2)=(f(x)+f(y))/2 for all real x and y. If f'(0) exists and equals -1 and f(0)=1, find f(2)

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((x)/(y))=(f(x))/(f(y)) for all x, y in R, y ne 0 and f'(x) exists for all x, f(2) = 4. Then, f(5) is

A function f : R rarr R satisfies the equation f(x + y) = f(x) . f(y) for all, f(x) ne 0 . Suppose that the function is differentiable at x = 0 and f'(0) = 2. Then,

A function f : R to R satisfies the equation f(x+y) = f (x) f(y), AA x, y in R and f (x) ne 0 for any x in R . Let the function be differentiable at x = 0 and f'(0) = 2. Show that f'(x) = 2 f(x), AA x in R. Hence, determine f(x)

Consider function f(x) satisfies the relation f(x+y^(3))=f(x)+f(y^(3))AAx,yinRand differentiable for all x . Statement I If f'(2)=a then f'(-2)=a f(x) is an odd function.

Let f be a function satisfying f(x+y)=f(x) + f(y) for all x,y in R . If f (1)= k then f(n), n in N is equal to