第7章 7.18

$f_{(x,y)} = \begin{vmatrix} \sin x & 0 & \sin y \\ 1 & 1 & -\cos y \\ -\cos x & 1 & 1 \end{vmatrix} \quad (0 \le x, y \le 2\pi)$

$= \sin x + \sin y + \cos x \cdot \sin y + \sin x \cdot \cos y$

$f_x = \cos x - \sin x \cdot \sin y + \cos x \cdot \cos y = \cos x + \cos(x+y)$

$f_y = \cos y + \cos x \cdot \cos y - \sin x \cdot \sin y = \cos y + \cos(x+y)$

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

$f_x = f_y = 0 \text{ のとき} \quad (x,y) = \left(\pm\frac{\pi}{3}, \pm\frac{\pi}{3}\right)$

$H_{\left(\pm\frac{\pi}{3},\,\pm\frac{\pi}{3}\right)} > 0$

$f_{xx\left(\frac{\pi}{3},\frac{\pi}{3}\right)} < 0 \qquad \text{極大} \qquad f_{xx\left(-\frac{\pi}{3},-\frac{\pi}{3}\right)} > 0 \qquad \text{極小}$

よって $\left(\dfrac{\pi}{3}, \dfrac{\pi}{3}\right)$ のとき 極大値 $\dfrac{3}{2}\sqrt{3}$

$\left(-\dfrac{\pi}{3}, -\dfrac{\pi}{3}\right)$ のとき 極小値 $-\dfrac{3}{2}\sqrt{3}$