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?

Jun 18, 2018

FALSE.

Counter-example:
$f \left(x\right) = {x}^{2} + 3 x + 2$ and $g \left(x\right) = {x}^{2} + 3 x + 5$

The derivatives of both functions are the same:
$f ' \left(x\right) = g ' \left(x\right) = 2 x + 3$
and yet, $f \left(x\right) \ne g \left(x\right)$

Explanation:

Given a derivative $f ' \left(x\right)$, you can apply the anti-derivative to obtain "a version" of the original function $f \left(x\right)$, 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 $f ' \left(x\right) = 2 x + 3$
then, the anti-derivative is:
$f \left(x\right) = {x}^{2} + 3 x + k$
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 $f \left(x\right) = {x}^{2} + 3 x + 2$ and $g \left(x\right) = {x}^{2} + 3 x + 5$

The derivatives of both functions are the same:
$f ' \left(x\right) = g ' \left(x\right) = 2 x + 3$
and yet, $f \left(x\right) \ne g \left(x\right)$