If f‘(x) = g‘(x), then f(x) = g(x). Explain your answer if it is true. If false, provide a counterexample. True or False?

1 Answer
Jun 18, 2018

FALSE.

Counter-example:
f(x)=x^2+3x+2 and g(x)=x^2+3x+5

The derivatives of both functions are the same:
f'(x) = g'(x) = 2x+3
and yet, f(x) != g(x)

Explanation:

Given a derivative f'(x), you can apply the anti-derivative to obtain "a version" of the original function f(x), but not the exact function, since there can always be a constant k added to the end, and that constant can have any value.

For example, if f'(x)=2x+3
then, the anti-derivative is:
f(x)=x^2+3x+k
where k can have any value.
So, there is no guarantee that you will arrive to the same function every time, and the statement in question is FALSE.

Here's a counter-example:
Let f(x)=x^2+3x+2 and g(x)=x^2+3x+5

The derivatives of both functions are the same:
f'(x) = g'(x) = 2x+3
and yet, f(x) != g(x)