题目
8.设f(x,y,z)=xy^2+yz^2+zx^2,求f_(xx)(0,0,1),f_(xz)(1,0,2),f_(yz)(0,-1,0)及f_(mx)(2,0,1).
8.设$f(x,y,z)=xy^{2}+yz^{2}+zx^{2}$,求$f_{xx}(0,0,1)$,$f_{xz}(1,0,2)$,$f_{yz}(0,-1,0)$及$f_{mx}(2,0,1)$.
题目解答
答案
计算一阶偏导数：
\[
f_x = y^2 + 2zx, \quad f_y = 2xy + z^2, \quad f_z = 2yz + x^2
\]
计算二阶偏导数：
\[
f_{xx} = 2z, \quad f_{xz} = 2x, \quad f_{yz} = 2z
\]
计算三阶偏导数：
\[
f_{zz} = 2y, \quad f_{zzx} = 0
\]
在给定点处求值：
\[
f_{xx}(0,0,1) = 2, \quad f_{xz}(1,0,2) = 2, \quad f_{yz}(0,-1,0) = 0, \quad f_{zzx}(2,0,1) = 0
\]
**答案：**
\[
\boxed{
\begin{array}{ll}
f_{xx}(0,0,1) = 2, \\
f_{xz}(1,0,2) = 2, \\
f_{yz}(0,-1,0) = 0, \\
f_{zzx}(2,0,1) = 0.
\end{array}
}
\]