8. Consider the function $f(x) = \frac{x^T M x}{x^T x}$ for $x \in \mathbb{R}^n$, $x \neq 0$ where $M$ is an $n \times n$ P.D. matrix. Prove that the global minima and maxima of this function are $\lambda$ and $\overline{\lambda}$ (the smallest and largest eigenvalues of $M$), respectively. Find the $x$'s corresponding to these extrema.
9. Compute the derivative $Df$ of the following functions.
(a) $f(x, y) = 3x^2 + 4xy + y^3$, $x, y \in \mathbb{R}$
(b) $f(x, y) = \begin{bmatrix} x^2 + 2y \ x + y^2 \end{bmatrix}$, $x, y \in \mathbb{R}$
(c) $f(x) = (Ax + B)^T M (Ax + B)$, $M = M^T$, $A, M \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^n$