I have a real analysis question?

Question

If f"(x)=0 everywhere on the interval (0,1) then f(x)=c , (c is a constant) on (0.1) does anyone have any suggetions on how to go about solving this

Answer 1

Use the mean value theorem.
Suppose f(a) and f(b) aren't equal for some a,b in (0,1).
By the mean value theorem, there is c in (0,1) with f'(c) = (f(b) - f(a))/(b-a).
But f'(c) = 0, so f(b) = f(a), contradiction.

Answer 2

If the derivative is zero, it means the slope is zero everywhere on the interval. Integrate the slope and get the solution that says the equation represents a straight line which is equal to a constant.