kaoyan2advanced 高等数学 第142题
📝 题目
### 第142题
设函数 $f(x, y)$ 在 $M_{0}\left(x_{0}, y_{0}\right)$ 处取极大值，且 $\displaystyle \left.\frac{\partial^{2} f}{\partial x^{2}}\right|_{M_{0}}$ 与 $\displaystyle \left.\frac{\partial^{2} f}{\partial y^{2}}\right|_{M_{0}}$ 存在，则 （A）$\displaystyle \left.\frac{\partial^{2} f}{\partial x^{2}}\right|_{M_{0}} \geqslant 0,\left.\frac{\partial^{2} f}{\partial y^{2}}\right|_{M_{0}} \geqslant 0$. （B）$\displaystyle \left.\frac{\partial^{2} f}{\partial x^{2}}\right|_{M_{0}} \leqslant 0,\left.\frac{\partial^{2} f}{\partial y^{2}}\right|_{M_{0}} \leqslant 0$. （C）$\displaystyle \left.\frac{\partial^{2} f}{\partial x^{2}}\right|_{M_{0}} \geqslant 0,\left.\frac{\partial^{2} f}{\partial y^{2}}\right|_{M_{0}} \leqslant 0$ ． （D）$\displaystyle \left.\frac{\partial^{2} f}{\partial x^{2}}\right|_{M_{0}} \leqslant 0,\left.\frac{\partial^{2} f}{\partial y^{2}}\right|_{M_{0}} \geqslant 0$ ．
💡 答案解析
**答案**：B **解析**：函数$f(x,y)$在$M_0$处取极大值，则固定$y=y_0$时，$f(x,y_0)$在$x=x_0$处取极大值，故$\displaystyle \left.\frac{\partial^2 f}{\partial x^2}\right|_{M_0}\leq 0$；同理，固定$x=x_0$时，$f(x_0,y)$在$y=y_0$处取极大值，故$\displaystyle \left.\frac{\partial^2 f}{\partial y^2}\right|_{M_0}\leq 0$。 **难度**：★★☆☆☆