Antiderivative Proof

[PaulKraemer]

Hi,


I am stuck on the following problem...

Suppose F and G are antiderivatives of f and g, respectively. Prove or disprove the statement that FG is an antiderivative of fg.

... What I have done so far is...

(1) I know that because F is an antiderivative of f, this means that F' = f
(2) I know that because G is an antiderivative of g, this means that G' = g
(3) I let H = FG and let h = fg. To prove thatt H is an antiderivative of h, I would have to show that H' = h (= fg)
(4) Calculating H' using the product rule, I get:

H' = F'G + G'F
H' = fG + gF

Maybe I'm wrong, but it seems to me that it is unlikely that fG + gF would be equal to fg for all possible functions F, G, f & g. This would mean that the statement that I am trying to prove or disprove would be false. I just can't figure out how to prove this.

Should I pick a some sample functions f and g with there corrsponding antiderivatives and try to find one set of functions where fG + fF <> fg.

Or is there a way that I can show this for all possible functions f, g, F, and G?

Thanks in advance,
Paul

 

[pka]

Suppose F and G are antiderivatives of f and g, respectively. Prove or disprove the statement that FG is an antiderivative of fg.

Give a counterexample: \(\displaystyle f(x)=2x~\&~g(x)=3x^2.\)