(6.0分)下列函数在(0.0)点处取得极大值的是:( )A:=2(x)^2+3(y)^2;B:=2(x)^2+3(y)^2 ;C:=2(x)^2+3(y)^2 . D:=2(x)^2+3(y)^2;

0, \frac{\partial^{2} z}{\partial x^{2}} \cdot \frac{\partial^{2} z}{\partial y^{2}}-\left(\frac{\partial^{2} z}{\partial x \partial y}\right)^{2}>0,(0,0) 为极小值点\\ &B : 由 z=-\sqrt{x^{2}+y^{2}} 可知 z 值域为 (-\infty, 0] ，\\ &当 (x, y) \neq(0,0) 时,z=-\sqrt{x^{2}+y^{2}}<0, \\ &只有当 (x, y)=(0,0) 时, z=0,(0,0) 为极大值点\\ &C: 令 \frac{\partial z}{\partial x}=y=0, \frac{\partial z}{\partial y}=x=0\\ &解得 x=0, y=0\\ &\frac{\partial^{2} z}{\partial x^{2}}=0, \quad \frac{\partial^{2} z}{\partial x y}=1, \frac{\partial^{2} z}{\partial y^{2}}=0\\ &\frac{\partial^{2} z}{\partial x^{2}} \frac{\partial^{2} z}{\partial y^{2}}-\left(\frac{\partial^{2} z}{\partial x x^{2}}\right)^{2}=-1<0, (0,0) 点不是极值点\\ &D. 令 \frac{\partial z}{\partial x}=-2 x=0, \frac{\partial z}{\partial y}=2 y=0\\ &解得 x=0, y=0\\ &\frac{\partial^{2} z}{\partial x^{2}}=-2, \frac{\partial^{2} z}{\partial x^{2} y}=0, \frac{\partial^{2} z}{\partial y^{2}}=2\\ &\frac{\partial^{2} z}{\partial x^{2}} \cdot \frac{\partial^{2} z}{\partial y^{2}}-\left(\frac{\partial^{2} z}{\partial x \partial y}\right)^{2}=-4<0 ，(0,0) 点不是极值点\\ &选B \end{aligned}" data-width="518" data-height="753" data-size="77710" data-format="png" style="max-width:100%">