Suppose $A$ is a symmetric positive definite matrix of size $nn$ with eigenvalues arranged in descending order. Prove the following inequality for all vectors $uin{R}^n$ where $||u|{|}_2=1$ :

$2\sqrt[\phantom{2}]{u^{T}Au}\sqrt[\phantom{2}]{u^{T}A{A}^{-1}u}\sqrt[\phantom{2}]{\frac{{}_1}{{}_n}}+\sqrt[\phantom{2}]{\frac{{}_n}{{}_1}}$