Difference between revisions of "Homogenization"
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==Solution== | ==Solution== | ||
| − | So all the terms except for the <math>1</math> are of the second degree. We substituting | + | So all the terms except for the <math>1</math> are of the second degree. We substituting <math>a+b+c</math> for <math>1</math>. The inequality still gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get <cmath>a^2+b^2+c^2+2(ab+bc+ca)=1</cmath> Now plugging this for <math>1</math> in the inequality and simplifying gives <math>a^2+b^2+c^2\ge ab+bc+ca</math>, which is well-known by |
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| − | <math>a+b+c</math> for <math>1</math>. The inequality still gives a non-homogeneous inequality. | ||
| − | |||
| − | So instead we square the condition to make it second degree and get <cmath>a^2+b^2+c^2+2(ab+bc+ca)=1</cmath> Now plugging this for <math>1</math> in the inequality and simplifying gives <math>a^2+b^2+c^2\ge ab+bc+ca</math>, which is well-known by | ||
Cauchy-Schwarz Inequality. | Cauchy-Schwarz Inequality. | ||
| − | We can use homogenization to help us solve these types of problems, especially inequalities. | + | We can use homogenization to help us solve these types of problems, especially inequalities. It's use is not limited. After making something homogenous we can often apply well known inequalities to solve problems. |
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| − | It's use is not limited. After making something homogenous we can often apply well known inequalities | ||
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| − | |||
{{stub}} | {{stub}} | ||
[[Category:Inequality]] | [[Category:Inequality]] | ||
Revision as of 23:01, 23 January 2014
Homogenizing is a useful technique to solve certain multivariate inequalities. Given an inequality of the form 
, where 
is a homogenous polynomial (that is, the degree of any term in the polynomial is the same), then we can arbitrarily impose a restraint of one order.
Example
If 
and 
, prove that 
.
Solution
So all the terms except for the 
are of the second degree. We substituting 
for . The inequality still gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get ![\[a^2+b^2+c^2+2(ab+bc+ca)=1\]](http://latex.artofproblemsolving.com/3/8/c/38cfebd204c87b71364175316460042e15319f32.png)
Now plugging this for in the inequality and simplifying gives 
, which is well-known by
Cauchy-Schwarz Inequality.
We can use homogenization to help us solve these types of problems, especially inequalities. It's use is not limited. After making something homogenous we can often apply well known inequalities to solve problems.
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