Why $a'(X'X)^{-1}a\leq \lambda_{\max}[(X'X)^{-1}]$ with $a'a=1$ and $\lambda_{\max}[\cdot]$ gives the largest eigenvalue of a matrix?

Question

In a lecture note I read about the following line in a proof: $a'(X'X)^{-1}a\leq \lambda_{\max}[(X'X)^{-1}]=[\lambda_{\min}(X'X)]^{-1}$, where $a'a=1$, $\lambda_{\max}[(X'X)^{-1}]$ gives the largest eigenvalue of matrix $(X'X)^{-1}$ and $\lambda_{\min}(X'X)$ gives the minimum eigenvalue of matrix $X'X$. I'm wondering why $a'(X'X)^{-1}a\leq \lambda_{\max}[(X'X)^{-1}]$ and why $\lambda_{\max}[(X'X)^{-1}]=[\lambda_{\min}(X'X)]^{-1}$. Both seems quite unintuitive to me, and it would be great if you could provide the theorems or results used here. Thanks!