I don't know how to do this problem! Deals with optimization, geometry, and modeling

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marchbat

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This is the only problem on my homework I don't know how to do! I would be so happy if someone were to help me.

The arithmetic mean of two numbers a and b is defined as (a+b)/2; the geometric mean of two positive numbers a and b is defined as (ab)^1/2.
(a) For two positive numbers, which of the two means is larger? Justify your answer. [Hint: Define f(x) = (a + x)/2 - (ax)^1/2.}

(b) For three positive numbers a, b, c the arithmetic and geometric mean are (a+b+c)/3 and (abc)^1/3, respectively. Which of the two means of three numbers is larger? [Hint: Redefine f(x) for fixed a and b.]

Edit: I was informed that I should be clear on what specifically I need help with, but I honestly don't know where to get started! The only thing I can think of is that if I get a positive number than (a+b)/2 is larger and if I get a negative number (ab)^1/2 is larger. Not sure if I'm on the right track PLEASE HELP.
 
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Denis said:
Dana, there is nothing preventing YOU from being part of "They".
You really could have simply said:
1. Have you tried anything yet?
2. Read the 'read before posting' thread.
Click to expand...

I put an edit under the problem with what I have tried or thought about. Could someone please help?
 
marchbat said:
I put an edit under the problem with what I have tried or thought about. Could someone please help?
Click to expand...

The way the problem (a) is stated it looks like they want you to find the minimum(s) and maximum(s) of the function
f(x) = (a + x)/2 - (ax)^1/2.
That is you have two numbers a and x and the function f(x) is defined by the arithmetic mean minus the geometric mean. Now, if you could say, for example, that f(x) was always non-negative, f(x) \(\displaystyle \ge\) 0, then you could say the arithmetic mean was always larger than or equal to the geometric mean.

Edit to add: One might ask how one can write the function that way when we want to prove the statement for any two numbers A and B. Well, for any two numbers A & B, we can let a = A and x = B. You can use the same technique on (b) as the hint indicates.
 
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