Problem 1
By direct matrix multiplication, express each of the following matrix products as a quadratic form: \(\left(\begin{array}{lll}a & {\left[\begin{array}{ll}u & v\end{array}\right]\left[\begin{array}{ll}4 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{l}u \\\ v\end{array}\right]}\end{array}\right]\) \((b)\left[\begin{array}{ll}u & v\end{array}\right]\left[\begin{array}{rr}-2 & 3 \\ 1 & -4\end{array}\right]\left[\begin{array}{l}u \\ v\end{array}\right]\) \((c)\left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{ll}5 & 2 \\\ 4 & 0\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]\) \((d)[d x \quad d y]\left[\begin{array}{ll}f_{x x} & f_{x y} \\ f_{y x} & f_{y y}\end{array}\right]\left[\begin{array}{l}d x \\ d y\end{array}\right]\)
Problem 1
Find the extreme values, if any, of the following four functions. Check whether they are maxima or minima by the determinantal test. $$z=x_{1}^{2}+3 x_{2}^{2}-3 x_{1} x_{2}+4 x_{2} x_{3}+6 x_{3}^{2}$$
Problem 2
Find the extreme values, if any, of the following four functions. Check whether they are maxima or minima by the determinantal test. $$z=29-\left(x_{1}^{2}-x_{2}^{2}+x_{3}^{2}\right)$$
Problem 2
A two-product firm faces the following demand and cost functions: \(Q_{1}=40-2 P_{1}-P_{2} \quad Q_{2}=35-P_{1}-P_{2} \quad C=Q_{1}^{2}+2 Q_{2}^{2}+10\) (a) Find the output levels that satisfy the first-order condition for maximum profit. (Use fractions.) (b) Check the second-order sufficient condition. Can you conclude that this problem possesses a unique absolute maximum? (c) What is the maximal profit?
Problem 4
Do the following constitute convex sets in the 3 -space? (a) A doughnut (b) A bowling pin (c) A perfect marble
Problem 4
Express each of the following quadratic forms as a matrix product involving a symmetric coefficient matrix: (a) \(q=3 u^{2}-4 u v+7 v^{2}\) (b) \(q=u^{2}+7 u v+3 v^{2}\) (c) \(q=8 u v-u^{2}-31 v^{2}\) (d) \(q=6 x y-5 y^{2}-2 x^{2}\) (e) \(q=3 u_{1}^{2}-2 u_{1} u_{2}+4 u_{1} u_{3}+5 u_{2}^{2}+4 u_{3}^{2}-2 u_{2} u_{3}\) (f) \(q=-u^{2}+4 u v-6 u w-4 v^{2}-7 w^{2}\)
Problem 4
Find the extreme values, if any, of the following four functions. Check whether they are maxima or minima by the determinantal test. $$z=e^{2 x}+e^{-y}+e^{w^{2}}-\left(2 x+2 e^{w}-y\right)$$
Problem 5
The equation \(x^{2}+y^{2}=4\) represents a circle with center at (0,0) and with a radius of 2 (a) Interpret geometrically the set \(\left.f(x, y) | x^{2}+y^{2} \leq 4\right\\}\) (b) Is this set convex?
Problem 6
Graph each of the following sets, and indicate whether it is convex: (a) \(\left\\{(x, y) | y=e^{x}\right\\}\) (c) \(\left\\{(x, y) | y \leq 13-x^{2}\right\\}\) (b) \(\left\\{(x, y) | y \geq e^{x}\right\\}\) (d)\(\\{(x, y) | x y \geq 1 ; x>0, y>0\\}\)
Problem 7
Given \(u=\left[\begin{array}{r}10 \\ 6\end{array}\right]\) and \(v=\left[\begin{array}{l}4 \\ 8\end{array}\right],\) which of the following are convex combinations of \(u\) \((a)\left[\begin{array}{l}7 \\ 7\end{array}\right]\) \((b)\left[\begin{array}{l}5.2 \\ 7.6\end{array}\right]\) \((c)\left[\begin{array}{l}6.2 \\ 8.2\end{array}\right]\)
