4.9 Lagrange's multiplier

1.

(a) at $x = \frac{\sqrt{2}}{2}$ , $y = -2\sqrt{2}$ is local minimum，at $x = -\frac{\sqrt{2}}{2}$ , $y = 2\sqrt{2}$ is local maximum．

(b) at $x = 1$ , $y = -\frac{1}{2}$ is local maximum，at $x = -1$ , $y = \frac{1}{2}$ is local minimum．

(c) at $x = 2 \sqrt[3]{2}$ , $y = 2\cdot2^{\frac{2}{3}}$ is local minimum．

2.

(a) maximum $\displaystyle{-\frac{3\sqrt{3}}{16}}$ , maximum $\displaystyle{\frac{3\sqrt{3}}{16}}$ (b) at $(0,0)$ , minimum value 0, at $(3,3)$ , maximum value $18$

(c) at $(\pm 1, \pm 1)$ , maximum value 1, at $\displaystyle{\pm \frac{1}{\sqrt{3}}, \mp \frac{1}{\sqrt{3}}}$ , minimum value $\displaystyle{-\frac{1}{3}}$

3.

$6$

4.

at $(0,0,1)$ , maximum value 3， at $(1,0,0)$ minimum value 1