Expert: Paul Klarreich - 5/11/2008

Question
1) Let: Z(x)=F(x)*G(x)
       F(x)>0 and G(x)>0
       F'(x) and G'(x) exist
       F"(x) and G"(x) exist

So Prove: If F(x) and G(x) both have a relative maximum at x=a then Z(x)has a relative maximum at x=a

Answer
Questioner:   David
Category:  Calculus
Private:  No
 
Subject:  relative max & min problem
Question:  1) Let: Z(x)=F(x)*G(x)
      F(x)>0 and G(x)>0
      F'(x) and G'(x) exist
      F"(x) and G"(x) exist

Prove: If F(x) and G(x) both have a relative maximum at x=a then Z(x)has a relative maximum at x=a
................................
Hi, David,

You are expected to draw some conclusions, such as:

If F(x) is positive, and G(x) is positive, then Z(x) is ?????1

IF F and G are both twice differentiable, then their product is ?????2

If F has a relative max at  x = a, and F is twice differentiable, then  F'(a) is ?????3, and F''(a) is ?????4

Did you say:
1. positive.
2. twice differentiable.
3. zero.
4. negative.
....................
Then you did well.

Now what about Z(a), Z'(a), Z''(a).

Z' = FG' + F'G by the product rule, and so

Z'(a) = F(a)G'(a) + F'(a)G(a)
     = F(a)(0) +  (0)G(a)
     = 0
So Z has a ?????5 at x = a.

Z'' = fg'' + f'g' + f'g' + f''g

Sorry -- I'm getting tired of holding down the shift key.

At x = a, you have:

Z'' = f(a)g''(a) + f'(a)g'(a) + f'(a)g'(a) + f''(a)g(a)

which is:

Z'' = (pos)(neg) + (0)(0) + (0)(0) + (neg)(pos)
   = negative

So Z has a ????6 at x = a.

Did you say

5. stationary point.
6. maximum.

Then you did well, and you have your proof.

================ FOLLOWUP.==================
OK, here is the outline of what you will write:

to prove Z has a rel. max. at  x = a, you want to prove three things about Z(x):

1. It is twice differentiable.
2. Z'(a) = 0
3. Z''(a) is negative.

I think the above work shows you how.