2022TrialShort answer

Q13 (c) (iii)

1 mark

Question

Hence prove for all $a_1,a_2>0$ that
$(a_1+a_2)\left(\frac1{a_1}+\frac1{a_2}\right)\ge4.$

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Solution

(iii)

(a_1+a_2)\left(\frac1{a_1}+\frac1{a_2}\right)

=1+\frac{a_1}{a_2}+\frac{a_2}{a_1}+1

\ge 1+2+1

=4\quad\left[\text{as }\frac{a_1}{a_2}+\frac{a_2}{a_1}\ge2\text{ by (ii)}\right].

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Proof

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

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