Prove the statement If f'(x) = g'(x) for all x and f(5)=g(5), then f(x) = g(x) for all x for all x0
Use one of the following: the mean value theorem, Rolle's theorem, the increasing function theorem, the constant function theorem, or the racetrack principle.

Question

Prove the statement If f'(x) = g'(x) for all x and f(5)=g(5), then f(x) = g(x) for all x for all x>0
Use one of the following: the mean value theorem, Rolle's theorem, the increasing function theorem, the constant function theorem, or the racetrack principle.

anonymous · Answer

if 
$$f'(x)=g'(x)$$ then 
$$f(x)=g(x)+c$$ and 
$$f(5)=g(5)\implies c = 0$$

anonymous · Answer

oh i see what you need, you need the first line that i wrote, that is
$$f'(x)=g'(x)\implies f(x)=g(x)+c$$ we need to prove that one

anonymous · Answer

suppose 
$$h'(x)=0$$ for all x, then by the mean value theorem 
$$\frac{h(x)-h(y)}{x-y}=0$$ for all pairs x and y
that means for all pairs x and y 
$$h(x)=h(y)$$ showing that 
$$h(x)=c$$ a constant

anonymous · Answer

apply this to the function 
$$h(x)=f(x)-g(x)$$ since 
$$f'(x)=g'(x)$$ this means that 
$$h'(x)=f'(x)-g'(x)=0$$ for all x, meaning that 
$$h(x)=f(x)-g(x)=c$$ a constant and so 
$$f(x)=g(x)+c$$ for some constant c

anonymous · Answer

now  c is a constant (does not depend on x) we know that 
$$f(5)=g(5)+c$$ and we also are given that 
$$f(5)=g(5)$$ we know that 
$$c=0$$