How Do You Prove a < sqrt(ab) < (a+b)/2 < b?

[monolithic]

Homework Statement

0 < a < b

Must prove that a < sqrt(ab) < (a+b)/2 < b

Homework Equations

Well, relevant info. I am using Spivak's calculus book, where we have to prove everything.

The Attempt at a Solution

I'm not sure if I'm doing this right but here's my start.

I'll start with a < b. Multiply both sides by a, and you get a times a < a times b.

Then you get a^2 < ab

Then square root of a^2 < square root of ab.

a < square root of ab.

So I think i have the first part of that, but I'm not sure where to start for a+b/2?

 

[VietDao29]

Homework Statement

0 < a < b

Must prove that a < sqrt(ab) < (a+b)/2 < b

Homework Equations

Well, relevant info. I am using Spivak's calculus book, where we have to prove everything.

The Attempt at a Solution

I'm not sure if I'm doing this right but here's my start.

I'll start with a < b. Multiply both sides by a, and you get a times a < a times b.

Then you get a^2 < ab

Then square root of a^2 < square root of ab.

a < square root of ab.

So I think i have the first part of that, but I'm not sure where to start for a+b/2?

$$\frac{a + b}{2} \geq \sqrt{ab}$$

This is actually a special case of http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means#The_inequality" (the case for n = 2, 2 terms). This case (n = 2) can be easily proven by isolating everything to one side, and use the fact that:

$$(x - y) ^ 2 = x ^ 2 - 2xy + y ^ 2 \geq 0$$

And for $$x \neq y$$, we have the equality:

$$(x - y) ^ 2 = x ^ 2 - 2xy + y ^ 2 {\color{red}>} 0$$

Let's see if you can take it from here.

And the last equality should be easy. :)

 

[njama]

VietDao29 did you mean:

$$(\sqrt{x}-\sqrt{y})^2 \geq 0$$ ?

 

[VietDao29]

VietDao29 did you mean:

$$(\sqrt{x}-\sqrt{y})^2 \geq 0$$ ?

No, I mean x, y; x, y in general.

 

[elduderino]

My post got deleted?! I am new to the forum here.. someone please shed light.

 

[njama]

No, I mean x, y; x, y in general.

If he use x=a, and y=b in the inequality that I posted, it will be very easy to prove it.

 

[HallsofIvy]

My post got deleted?! I am new to the forum here.. someone please shed light.

You gave a complete solution rather than helping hints.