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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?

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Jun 18, 2018

Answer:

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)#

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