91Ó°ÊÓ

Find necessary and sufficient conditions for a \(2 \times 2\) symmetric matrix to be negative definite. Use this information to state and prove a sufficient condition for a point to be a local maximizer for a function of two variables.

Show that the point (-1,1) is the minimizer of the function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) defined by $$ f(x, y)=(2 x+3 y)^{2}+(x+y-1)^{2}+(x+2 y-2)^{2} \quad \text { for }(x, y) \text { in } \mathbb{R}^{2} $$

Analyze the local extreme points of the function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) defined by $$ f(x, y)=\cos (x+y)+\sin (x+y) \quad \text { for }(x, y) \text { in } \mathbb{R}^{2} $$

Suppose that the continuous function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) has a tangent plane at the point \(\left(x_{0}, y_{0}, f\left(x_{0}, y_{0}\right)\right) .\) Prove that the function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) has directional derivatives in all directions at the point \(\left(x_{0}, y_{0}\right)\)