Math Is Fun Forum

pineapple12

Member

Registered: 2016-06-21

Posts: 8

Algebraic Inequalities

Hi,

Prove that

sqrt((2x^2 - 2x + 1) / 2) >=1/(x + 1/x)

for $0 < x < 1.$

This is the original problem. I'm trying to prove an intermediate inequality sqrt((2x^2 - 2x + 1) / 2) >= 1/2 >= 1/(x+1/x) .

I used AM-GM on x + 1/x, and I ended up with 1/(x + 1/x) <= 1/2.
Now, I'm kind of having trouble finding a way to prove that sqrt((2x^2 - 2x + 1) / 2) >= 1/2. I don't know if I should use a mean inequality or what to do. I'm so close to solving this problem, so any strong hints would be appreciated.

Thank you so much!!!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebraic Inequalities

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebraic Inequalities

When stuck try calculus:

Easy to minimize that to see that the minimium occurs at x = 1 / 2 and is equal to - 1 / 4

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebraic Inequalities

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

pineapple12

Member

Registered: 2016-06-21

Posts: 8

Re: Algebraic Inequalities

Wow! Oh my gosh thank you thank you thank you thank you so much!!! You guys did more than I expected, thank you guys!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebraic Inequalities

Okay, good luck.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.