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

  • Thread starter monolithic
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Please give hints instead.In summary, the problem is to prove the inequality a < sqrt(ab) < (a+b)/2 < b using Spivak's calculus book. To start, you can 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 > 0. Can you take it from here to prove the last two parts of the inequality?
  • #1
monolithic
1
0

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?
 
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  • #2
monolithic said:

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?

[tex]\frac{a + b}{2} \geq \sqrt{ab}[/tex]

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:

[tex](x - y) ^ 2 = x ^ 2 - 2xy + y ^ 2 \geq 0[/tex]

And for [tex]x \neq y[/tex], we have the equality:

[tex](x - y) ^ 2 = x ^ 2 - 2xy + y ^ 2 {\color{red}>} 0[/tex]

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

And the last equality should be easy. :)
 
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  • #3
VietDao29 did you mean:

[tex](\sqrt{x}-\sqrt{y})^2 \geq 0 [/tex] ?
 
  • #4
njama said:
VietDao29 did you mean:

[tex](\sqrt{x}-\sqrt{y})^2 \geq 0 [/tex] ?

No, I mean x, y; x, y in general.
 
  • #5
My post got deleted?! I am new to the forum here.. someone please shed light.
 
  • #6
VietDao29 said:
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. :smile:
 
  • #7
elduderino said:
My post got deleted?! I am new to the forum here.. someone please shed light.
You gave a complete solution rather than helping hints.
 

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

How can I prove my hypothesis?

In order to prove your hypothesis, you will need to conduct a scientific experiment. This involves creating a testable prediction based on your hypothesis, designing and conducting the experiment, and analyzing the data collected. Through this process, you can determine if your hypothesis is supported or not.

What type of evidence do I need to prove something?

The type of evidence needed to prove something depends on the nature of the claim being made. In general, scientific evidence should be empirical, meaning it is based on observations and measurements. This evidence should also be replicable, meaning that others should be able to conduct the same experiment and get similar results.

How can I ensure the validity of my experiment?

There are several ways to ensure the validity of your experiment. One is to use a control group, which is a group that does not receive the treatment being tested. This allows for comparison to the experimental group. Additionally, you should use a large sample size to reduce the impact of individual variations. It is also important to eliminate any potential biases or confounding variables in the experiment.

What role does statistical analysis play in proving something?

Statistical analysis is a crucial component of proving something in science. It allows you to determine if the results of your experiment are due to chance or if they are statistically significant. This involves using mathematical techniques to analyze the data and calculate the probability of obtaining the results by chance.

How can I communicate my findings to prove something?

Communicating your findings is an important part of the scientific process. You can do this through publishing your results in a scientific journal or presenting them at a conference. It is important to clearly and accurately report your methods, results, and conclusions in order to provide evidence for your claim.

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