2022HSCShort answer

Q12 (a)

2 marks

Question

For real numbers $a,b\ge0$ , prove that
$\frac{a+b}{2}\ge\sqrt{ab}.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Chooses a suitable strategy	1

Sample answer:
$a,b\ge0\text{ so }\sqrt a,\sqrt b\text{ are real}$

(\sqrt a-\sqrt b)^2\ge0

$a-2\sqrt{ab}+b\ge0$
$a+b\ge2\sqrt{ab}$
$\frac{a+b}{2}\ge\sqrt{ab}$

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Syllabus alignment

Proof

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.

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