let and thus is constant and the conclusion follows.
Why you think it is false?
If f'(x)=g'(x) then both functions have at x the same slope. For example x=1.
f'(1) = 3 then g'(x) = 3 --> this tells you that both graphs have a slope of 3 at x=3
but the funtion itself can be different. Like f(1) = 2 and g(1) = 4, then the two curves would have the same slope at x=1 but are lying parallel to each other.
The same for an maximum or minimum point. If f(1)=0 and g(1) =0 then both functions have an max or min at x=1 but the maximum/minimum value of f(x)could be at (1,3) and the other of g(x) could be at (1,4)
The function values I used are just random examples.
Thats how I understood it. Hope it helps.