Arithmetic mean and geometric mean

Thread starter FISMAQUIM
Start date Oct 21, 2024
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Expression Minimum

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The discussion focuses on finding the minimum value of the expression [1/(1+ab)] + [1/(1+bc)] + [1/(1+ac)] given that a² + b² + c² = 3 for positive real numbers a, b, and c. It is noted that the inequality A² + B² + C² ≥ AB + AC + BC can be applied to derive insights into the problem. The minimum value of the expression is determined to be 3/2. Participants are reminded that only hints and guidance can be provided for homework-type questions, emphasizing the importance of showing work in problem-solving. The discussion underscores the relevance of inequalities in optimizing expressions involving positive real numbers.

Oct 21, 2024

FISMAQUIM

Homework Statement: Arithmetic mean and geometric mean

Relevant Equations: Arithmetic mean and geometric mean

If a, b, and c are positive real numbers and a² + b² + c² = 3, what is the minimum value of the expression [1/(1+ab)] + [1/(1+bc)] + [1+(1+ac )]?

Usage: A² + B² + C² ≥ AB+AC+BC

Answer: 3/2

Last edited: Oct 21, 2024

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Oct 21, 2024

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On any homework-type of question we are only allowed to give hints and guidance regarding the work that you show. You must show work on this problem.

PS. Your "Usage" statement for general A, B, C could be stronger.

Last edited: Oct 21, 2024

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