Exercise 179Algebra › InequalitiesHigh schoolAdded 31 Oct 2026

A classic inequality

#inequalities#AM-GM

Statement

For $a,b>0$ , prove that
$(a+b)\left(\frac1a+\frac1b\right)\ge 4,$
with equality if and only if $a=b$ .

Hint

Expand the product, then set $t=a/b$ .

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Expanding gives $(a+b)\left(\frac1a+\frac1b\right)=2+\dfrac ab+\dfrac ba$ .

Let $t=a/b>0$ . Then $t+\dfrac1t-2=\dfrac{(t-1)^2}{t}\ge0$ , so the product is at least $4$ , with equality iff $t=1$ , that is $a=b$ .