A community for students.
Here's the question you clicked on:
 0 viewing
 2 years ago
Fun Challenge
So here is the challenge. Prove that (a + b)/2 (the arithmetic mean) is greater than or equal to sqrt(ab) (the geometric mean) for any real numbers a and b!
 2 years ago
Fun Challenge So here is the challenge. Prove that (a + b)/2 (the arithmetic mean) is greater than or equal to sqrt(ab) (the geometric mean) for any real numbers a and b!

This Question is Closed

LogicalApple
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac{ a + b }{ 2 } \ge \sqrt{ab}\]

watchmath
 2 years ago
Best ResponseYou've already chosen the best response.1Squaring both sides and rearranging we have \((ab)^2\geq 0\) which is always true. BTW you need the condition that \(a,b\) are non negative though!!

Homeworksucks
 2 years ago
Best ResponseYou've already chosen the best response.05+6/2=11 sqrt(30)= 5.47722557505 11 is greater than 5.4 1+1.00001/2= 1.00005 1*1.00001=1.00001 sqrt(1.00001)= 1.00000499999 I haven't studied calculus yet but isn't something like as a+b approaches 2 at the very limit, it should be equal to sqrt(ab)?

LogicalApple
 2 years ago
Best ResponseYou've already chosen the best response.0I should probably say that no calculus is required :)

Homeworksucks
 2 years ago
Best ResponseYou've already chosen the best response.0Dammit, that probably wasn't the kind of proof you were looking for wasn't it?

watchmath
 2 years ago
Best ResponseYou've already chosen the best response.1a=2 b=2. Arithmetic mean is 2 , but geometric mean is 2. So AM < GM in this case. You need a and b to be positive

LogicalApple
 2 years ago
Best ResponseYou've already chosen the best response.0Good point! That's what I get for working out of a textbook, huh? Great counterexample. Ok let's assume a and b are positive then.

watchmath
 2 years ago
Best ResponseYou've already chosen the best response.1The proof is as I said in the first post

LogicalApple
 2 years ago
Best ResponseYou've already chosen the best response.0You were not very explicit but you got the same result I did. Magic?
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.