第7章 7.9

(1)

$z = x^2 - xy + y^2 - 3x + y - 2$

$z_x = 2x - y - 3 \qquad z_y = -x + 2y + 1 \qquad z_{xx} = 2 \qquad z_{yy} = 2 \qquad z_{xy} = -1$

$z_x = z_y = 0$ のとき $(x,y) = (\dfrac{5}{3}, \dfrac{1}{3})$

$H_{(x,y)} = 4 - 1 = 3 > 0$

$z_{xx} = 2 > 0$ 極小

よって$(\dfrac{5}{3}, \dfrac{1}{3})$のとき 極小値$-\dfrac{13}{3}$

(2)

$z = x^2 + xy + y^2 + y^3$

$z_x = 2x + y \qquad z_y = x + 2y + 3y^2 \qquad z_{xx} = 2 \qquad z_{yy} = 2 + 6y \qquad z_{xy} = 1$

$z_x = z_y = 0$ のとき $(x,y) = (0,0)$

$H = 2(2+6y) - 1 \qquad H_{(0,0)} = 4 - 1 = 3 > 0$

$z_{xx} = 2 > 0$ 極小

よって$(0,0)$で極小値$0$

(3)

$z = 4x^3 - y^3 + 3x^2y + 9y$

$z_x = 12x^2 + 6xy \qquad z_y = -3y^2 + 3x^2 + 9 \qquad z_{xx} = 24x + 6y \qquad z_{yy} = -6y \qquad z_{xy} = 6x$

$z_x = z_y = 0$ のとき $(x,y) = (\pm 1, \mp 2)$

$H_{(-1,2)} = -12 \cdot (-12) - (-6)^2 > 0$

$z_{xx(-1,2)} < 0$ 極大

$H_{(1,-2)} = 12 \cdot 12 - 6^2 > 0$

$z_{xx(1,-2)} > 0$ 極小

よって$(1,-2)$のとき極小値$-12$ $(-1,2)$のとき極大値$12$

(4)

$z = \sin x + \cos y + \sin(x+y) \qquad (0 \le x, y \le 2\pi)$

$z_x = \cos x + \cos(x+y) \qquad z_y = -\sin y + \cos(x+y)$

$z_{xx} = -\sin x - \sin(x+y) \qquad z_{xy} = -\sin(x+y) \qquad z_{yy} = -\cos y - \sin(x+y)$

$z_x = 0$、$z_y = 0$ のとき

$\cos x + \cos(x+y) = 0 \qquad -\sin y + \cos(x+y) = 0 \qquad -\cos x = \sin y = \cos(x+y)$

これから $(x,y) = (\dfrac{\pi}{2},0)$、$(\dfrac{\pi}{2},2\pi)$、$(\dfrac{7\pi}{6},\dfrac{2\pi}{3})$、$(\dfrac{11\pi}{6},\dfrac{4\pi}{3})$ が考えられる。

$H_{(x,y)} = \sin x \cdot \cos y + (\sin x + \cos y)\sin(x+y)$

$H_{(\frac{\pi}{2},0)} = 1+2>0 \qquad z_{xx} = -1-1<0 \qquad \text{極大}$

$H_{(\frac{\pi}{2},2\pi)} > 0 \qquad z_{xx}<0 \qquad \text{極大}$

$H_{(\frac{7\pi}{6},\frac{2\pi}{3})} = \dfrac{1}{4} > 0 \qquad z_{xx} = \dfrac{1}{2}+\dfrac{1}{2} > 0 \qquad \text{極小}$

$H_{(\frac{11\pi}{6},\frac{4\pi}{3})} > 0 \qquad z_{xx} > 0 \qquad \text{極小}$

よって $(\dfrac{\pi}{2},0)$、$(\dfrac{\pi}{2},2\pi)$ のとき極大値 $3$ $(\dfrac{7\pi}{6},\dfrac{2\pi}{3})$、$(\dfrac{11\pi}{6},\dfrac{4\pi}{3})$ のとき極小値 $-\dfrac{3}{2}$