Problem
ALG-B1-M07-P006 A fraction and its reciprocal
#6
★★☆☆☆ Level 2 of 5
For \(a,b>0\), prove that \(\frac{a}{b}+\frac{b}{a}\ge2\).
Apply AM-GM to the two positive fractions.
By AM-GM, \(\frac{a}{b}+\frac{b}{a}\ge2\sqrt{\frac{a}{b}\cdot\frac{b}{a}}=2\). Equality holds when \(a=b\).
Preparation for normalizing ratios.