Problem
ALG-B1-M07-P008 A mixed fraction
#8
★★☆☆☆ Level 2 of 5
For \(a,b>0\), prove that \(\frac{x^2}{a}+\frac{y^2}{b}\ge\frac{(x+y)^2}{a+b}\) for all real \(x,y\).
This is exactly Cauchy in Engel form.
By Cauchy-Schwarz in Engel form, \(\frac{x^2}{a}+\frac{y^2}{b}\ge\frac{(x+y)^2}{a+b}\). Equality occurs when \(\frac{x}{a}=\frac{y}{b}\).
It is useful to state the equality condition explicitly.