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:R rarr 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 a real valued function f(x) satisfies the equation f(x+y)=f(x)+f(y) for all x,y in R then f(x) is

A function f:R rarr R satisfy the equation f(x)f(y)-f(xy)=x+y for all x,y in R and f(y)>0, then

Let f(x+y)=f(x)+f(y) for all real x,y and f'(0) exists.Prove that f'(x)=f'(0) for all x in R and 2f(x)=xf(2)

A function f:R rarr R satisfies that equation f(x+y)=f(x)f(y) for all x,y in R ,f(x)!=0. suppose that the function f(x) is differentiable at x=0 and f'(0)=2. Prove that f'(x)=2f(x)

If f((x + 2y)/3)=(f(x) + 2f(y))/3 AA x, y in R and f(x) is continuous at x = 0. Prove that f(x) is continuous for all x in R.

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' = 2f(x) .

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

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