f（x，xy）求二阶偏导数 求f（x，x／y）的二阶偏导数
\u4e8c\u9636\u504f\u5bfc\u6570\u600e\u4e48\u6c42\uff1f\u4e8c\u9636\u504f\u5bfc\u6570\u6c42\u6cd5\u4ecb\u7ecd\uff1a
\u8bbe\u6709\u4e8c\u5143\u51fd\u6570z=f(x,y)\uff0c\u70b9(x0,y0)\u662f\u5176\u5b9a\u4e49\u57dfD\u5185\u4e00\u70b9\u3002\u628ay\u56fa\u5b9a\u5728y0\u800c\u8ba9x\u5728x0\u6709\u589e\u91cf\u25b3x\uff0c\u76f8\u5e94\u5730\u51fd\u6570z=f(x,y)\u6709\u589e\u91cf\uff08\u79f0\u4e3a\u5bf9x\u7684\u504f\u589e\u91cf\uff09\u25b3z=f(x0+\u25b3x,y0)-f(x0,y0)\u3002
\u5982\u679c\u25b3z\u4e0e\u25b3x\u4e4b\u6bd4\u5f53\u25b3x\u21920\u65f6\u7684\u6781\u9650\u5b58\u5728\uff0c\u90a3\u4e48\u6b64\u6781\u9650\u503c\u79f0\u4e3a\u51fd\u6570z=f(x,y)\u5728(x0,y0)\u5904\u5bf9x\u7684\u504f\u5bfc\u6570\uff0c\u8bb0\u4f5cf'x(x0,y0)\u6216\u51fd\u6570z=f(x,y)\u5728(x0,y0)\u5904\u5bf9x\u7684\u504f\u5bfc\u6570\u3002
\u628ay\u56fa\u5b9a\u5728y0\u770b\u6210\u5e38\u6570\u540e\uff0c\u4e00\u5143\u51fd\u6570z=f(x,y0)\u5728x0\u5904\u7684\u5bfc\u6570\u3002\u540c\u6837\u628ax\u56fa\u5b9a\u5728x0\uff0c\u8ba9y\u6709\u589e\u91cf\u25b3y\uff0c\u5982\u679c\u6781\u9650\u5b58\u5728\u90a3\u4e48\u6b64\u6781\u9650\u79f0\u4e3a\u51fd\u6570z=(x,y)\u5728(x0,y0)\u5904\u5bf9y\u7684\u504f\u5bfc\u6570\u3002\u8bb0\u4f5cf'y(x0,y0)\u3002
\u516c\u5f0f\u4ecb\u7ecd\uff1a
1\u3001∂z/∂x=[\u221a(x²+y²)-x\u00b72x/2\u221a(x²+y²)]/(x²+y²)=y²/[(x²+y²)^(3/2)]
2\u3001∂z/∂y=-x\u00b72y/2\u221a(x²+y²)^(3/2)]=-xy/[(x²+y²)^(3/2)]
3\u3001∂²z/∂x²=-(3/2)y²\u00b72x/[(x²+y²)^(5/2)]=-3xy²/[(x²+y²)^(5/2)]
4\u3001∂²z/∂x∂y=[2y\u00b7[(x²+y²)^(3/2)-y²\u00b7(3/2)\u00b7[(x²+y²)^(1/2)2y]/[(x²+y²)³]
\u4e8c\u9636\u504f\u5bfc\u6570\u6027\u8d28\u4ecb\u7ecd\uff1a
\u4e00\u3001\u5982\u679c\u4e00\u4e2a\u51fd\u6570f(x)\u5728\u67d0\u4e2a\u533a\u95f4I\u4e0a\u6709f''(x)\uff08\u5373\u4e8c\u9636\u5bfc\u6570\uff09>0\u6052\u6210\u7acb\uff0c\u90a3\u4e48\u5bf9\u4e8e\u533a\u95f4I\u4e0a\u7684\u4efb\u610fx,y\uff0c\u603b\u6709\uff1af(x)+f(y)\u22652f[(x+y)/2]\uff0c\u5982\u679c\u603b\u6709f''(x)<0\u6210\u7acb\uff0c\u90a3\u4e48\u4e0a\u5f0f\u7684\u4e0d\u7b49\u53f7\u53cd\u5411\u3002
\u51e0\u4f55\u7684\u76f4\u89c2\u89e3\u91ca\uff1a\u5982\u679c\u4e00\u4e2a\u51fd\u6570f(x)\u5728\u67d0\u4e2a\u533a\u95f4I\u4e0a\u6709f''(x)\uff08\u5373\u4e8c\u9636\u5bfc\u6570\uff09>0\u6052\u6210\u7acb\uff0c\u90a3\u4e48\u5728\u533a\u95f4I\u4e0af(x)\u7684\u56fe\u8c61\u4e0a\u7684\u4efb\u610f\u4e24\u70b9\u8fde\u51fa\u7684\u4e00\u6761\u7ebf\u6bb5\uff0c\u8fd9\u4e24\u70b9\u4e4b\u95f4\u7684\u51fd\u6570\u56fe\u8c61\u90fd\u5728\u8be5\u7ebf\u6bb5\u7684\u4e0b\u65b9\uff0c\u53cd\u4e4b\u5728\u8be5\u7ebf\u6bb5\u7684\u4e0a\u65b9\u3002
\u4e8c\u3001\u8bbef(x)\u5728[a,b]\u4e0a\u8fde\u7eed\uff0c\u5728\uff08a,b)\u5185\u5177\u6709\u4e00\u9636\u548c\u4e8c\u9636\u5bfc\u6570\uff0c\u90a3\u4e48\uff1a
1\u3001\u82e5\u5728\uff08a,b)\u5185f''(x)>0,\u5219f(x)\u5728[a,b]\u4e0a\u7684\u56fe\u5f62\u662f\u51f9\u7684\uff1b
2\u3001\u82e5\u5728\uff08a,b)\u5185f\u2019\u2018(x)<0,\u5219f(x)\u5728[a,b]\u4e0a\u7684\u56fe\u5f62\u662f\u51f8\u7684\u3002
\u5bf9x\u6c42\u504f\u5bfc\u5f97\u5230
f'x=f1' +f2' *1/y
\u5bf9y\u6c42\u504f\u5bfc\u5f97\u5230
f'y=f2' *(-x/y^2)
\u4e8e\u662f\u6c42\u4e8c\u9636\u504f\u5bfc\u6570\u5f97\u5230
f''xx=f11'' +f12'' *1/y +(f21'' +f22'' *1/y) *1/y
f''xy=f12'' *(-x/y^2) -f2' *1/y^2 +f22'' *(-x/y^3)
f''yy=f22'' *x^2/y^4 +2f2' *x/y^3
方法如下，
请作参考：
方法如下，
请作参考：
如图，过程与结果如图所示
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