Quiz 22: Proving An Inequality

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Created by: Isabelle Yeong
Date: April 1st 2016

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Answer for Please answer the following questions based on proving the following inequality holds for all positive reals a, b and c such that abc=1:

Please answer the following questions based on proving the following inequality holds for all positive reals a, b and c such that abc=1:

Question 1: Would you see turning the RHS of the inequality of 1 as abc help?

A. Yes.

B. No.

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Question 2: If you're ask to apply the extended Cauchy-Schwarz inequality to the LHS of the intended inequality, what would be the first thing you would do to the LHS of the intended inequality?

A. Nothing, as we can directly apply the extended Cauchy-Schwarz inequality to the LHS of the intended inequality without a problem.

B. Multiply top and bottom of the fractions by $abc$.

C. Multiply the top and bottom of the fractions by whatever listed in the numerator of that particular fraction.

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Question 3: With the same condition as the original problem, let's say now you're ask to prove (a+b+c)²/(2(a+b+c)+3abc)≥ 1, what strategy would you plan?

A. Expand the numerator.

B. Replace (a+b+c)² by 3(ab+bc+ca) since (a+b+c)²≥3(ab+bc+ca).

C. Algebraically manipulate the given equality, which says abc=1 by letting a=x/y, b=y/z,z/x.

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Question 4: If you are encourage to prove this problem with the Hölder's inequality, do you think you can manage that and successfully prove the problem with just a few steps?

A. Yes.

B. No.

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