2008 iTest Problems/Problem 90
Revision as of 00:14, 31 December 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 90 (credit to official solution) -- Minimum of Three Variable Fraction)
Problem
For positive reals, let . Find the minimum value of .
Solution
By the Trivial Inequality (with equality happening if ), Add to both sides and use the reciprocal property to get Since , multiplying both sides by this value would not change the inequality sign, and doing so results in By using similar steps, we find that
Let , , and , making . Note that . By the Cauchy-Schwarz Inequality, , so . Equality happens if , which is possible if . If , then .
Therefore, the minimum value of (which happens if ) is , so the minimum value of is .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 89 |
Followed by: Problem 91 | |
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