使用拉格朗日乘法证明，当两数相等时乘积最大

采用拉格朗日乘法证明：当\(x=y=\frac{c}{2}\)时，\(xy\)取得最大值

已知，C是任意大于零的常数，且 x + y = c ，证明当 \(x=y=\frac{c}{2}\) 时，\(xy\) 取得最大值。

\[ \left\{ \begin{align} & f(x,y) = xy \\ & \phi(x,y) = x + y -c = 0 \end{align} \right. \]

先作拉格朗日函数

\[\begin{align} L(x,y) = f(x,y)+ \lambda\phi(x,y) \\ \end{align} \]

求得

\[ \left\{ \begin{align} & f_{x}(x,y) + \lambda\phi_{x}(x,y) = 0 \\ & f_{y}(x,y) + \lambda\phi_{y}(x,y) = 0 \\ & \phi(x,y) = x + y -c = 0 \end{align} \right. \]

即

\[ \left\{ \begin{align} & y + \lambda = 0 \\ & x + \lambda = 0 \\ & x + y = c \end{align} \right. \]

解得

\[ \left\{ \begin{align} & x = \frac{c}{2} \\ & y = \frac{c}{2} \end{align} \right. \]

证毕