Integration Proof with constant

[OmniNewton]

Homework Statement

Prove that if f'(x) = g'(x) for all x in an interval (a,b) then f-g is constant on (a,b) then f-g is constant on (a,b) that is f(x) = g(x) + C

Homework Equations

Let C be a constant
Let D be a constant

The Attempt at a Solution

f(x) = antiderivative(f'(x)) = f(x) + C
g(x)= antiderivative(g'(x)) = g(x) + D

f-g = f(x) + C - (g(x) + D)
f-g = f(x) - g(x) + C - D

but since f'(x) = g'(x) then f(x) = g(x) the only difference is their constant.

then,

f-g = f(x) - f(x) + C - D
f-g = C - D

Since C and D are constants

then,

f-g = constant

if C = D

then f-g = 0

Note: I feel like I proved it but my notation is wrong since I cannot use f(x) = f(x) + C. I would like guidance for the proper notation to use. The possibility also exists my proof is completely wrong. I would like help

Thanks in Advanced!

 

[Simon Bridge]

You have not used the interval.

 

[OmniNewton]

How would I got about approaching this problem? I'm sorry perhaps I am lacking knowledge but I have only received a 30 minute lesson on integrals in my class, which was the last class of the semester.

 

[PeroK]

How would I got about approaching this problem? I'm sorry perhaps I am lacking knowledge but I have only received a 30 minute lesson on integrals in my class, which was the last class of the semester.

Here's a suggestion to get you started. Let ##h(x) = f(x) - g(x)##. What can you say about ##h(x)##?

 

[OmniNewton]

Here's a suggestion to get you started. Let ##h(x) = f(x) - g(x)##. What can you say about ##h(x)##?

h(x) is constant on (a,b). Should I be using the area beneath the curve formula?

 

[lordianed]

h(x) is constant on (a,b). Should I be using the area beneath the curve formula?

How did you deduce that ##h(x)## is constant on ##(a,b)##? PeroK has a very good suggestion, under what conditions would ##h(x)## be constant over an interval? Could this have something to do with its derivative?