Quick questions on Proving inequalities in HSC Mathematics Extension 2: the non-negativity of squares, the AM-GM inequality, and rigorous algebraic argument

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is a geometric picture of AM-GM?

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The two-variable AM-GM result has a clean geometric meaning that is worth carrying in your head, because it instantly tells you why equality occurs only when

a = b

. Draw a semicircle on a diameter

PQ

of length

a + b

, and let

D

split the diameter into

PD = a

and

DQ = b

. Drop the perpendicular from

D

up to the circle at

C

What is algebraic proof?

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Since

a, b \ge 0

, both

\sqrt{a}

and

\sqrt{b}

are real. Start from the true statement

What is proof by the difference?

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Expand both sides and consider the difference

\text{RHS} - \text{LHS}

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