Two things: how do you find the slope of the tangent lines in general for f(x)? If a line is horizontal, what is its slope? How can you use these two things together to solve?
If that doesn't help I'll get you started.
The slope of a line that is tangent to a certain point on f(x) is different at different points because f(x) increases and decreases at different rates. Taking f'(a) will tell you the slope of a line that is tangent to the point (a, f(a)) on the graph of f(x), but the slope of the point (a+1) is going to be different.
You are right though that you need to find when f'(x)=0 and solve for x.