Question

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?

Answer 1

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)`