Weighted AM-GM inequality for JEE

Formula

\(\frac{\alpha x+\beta y}{\alpha+\beta}\ge (x^{\alpha}y^{\beta})^{1/(\alpha+\beta)}\)

Variables: \(x,y\): non-negative reals, \(\alpha,\beta\): positive weights.

Conditions: \(x,y\ge 0\), \(\alpha,\beta>0\); equality iff \(x=y\).

Weighted optimization and inequality transformations.