题目

1.设f(x,y)与φ(x,y)均为可微函数,且 varphi y(x,y)neq 0,-|||-已知(x0,y0)是f(x,y)在约束条件 varphi (x,y)=0-|||-下的一个极值点,下列选项正确的是(). ()-|||-A)若 _(x)((x)_(0),(y)_(0))=0, 则 _(y)((x)_(0),(y)_(0))=0-|||-(B)若 _(x)((x)_(0),(y)_(0))=0, 则 _(y)((x)_(0),(y)_(0))neq 0-|||-(C)若 _(x)((x)_(0),(y)_(0))neq 0, 则 _(y)((x)_(0),(y)_(0))=0-|||-(D)若 _(x)((x)_(0),(y)_(0))neq 0, 则 _(y)((x)_(0),(y)_(0))neq 0

1. $\begin{aligned} &\text{ 设 }f(x,y)\text{与}\varphi(x,y)\text{均为可微函数},\text{且 }\varphi_y(x,y)\neq0,\\&\text{已知}(x_0,y_0)\text{是 }f(x,y)\text{在约束条} \text{件 }\varphi(x,y)=0\\&\text{ 下的一个极值点,下列选项正确的是()}. \\ &\text{(A)若}f_x(x_0,y_0)=0,\text{则 }f_y(x_0,y_0)=0 \\ &\text{(B) 若 }f_x(x_0,y_0)=0,\text{则 }f_y(x_0,y_0)\neq0 \\ &(C)\text{ 若 }f_x(x_0,y_0)\neq0,\text{则 }f_y(x_0,y_0)=0 \\ &\text{(D) 若 }f_x(x_0,y_0)\neq0,\text{则 }f_y(x_0,y_0)\neq0 \end{aligned}$

题目解答

答案

解：

$\begin{aligned} &\mathbf{}\text{根据题意,可以构建拉格朗日函数:} \\ &F\left(x,y,\lambda\right)=f\left(x,y\right)+\lambda\varphi\left(x,y\right) \\ &M(x_{0}\textit{,}y_{0})\text{是}f(x,y)\text{在约束条件下}\varphi\left(x,y\right)=0\text{下的一个极} \\ &\text{值点,} \\ &根据拉格朗日函数极值点的条件,在 {(x_0\text{ ,}y_0)}\text{处要满}\text{足:} \\ &\left\{\begin{array}{l}F_x^{\prime}=0\\F_y^{\prime}=0\\F_{\lambda}^{\prime}=0\end{array}\right. \\ &\text{即:} \\ &F_{x}^{\prime}=f_{x}^{\prime}(x_{0},y_{0})+\lambda\varphi_{x}^{\prime}(x_{0},y_{0})=0; \\ &F_{y}^{\prime}=f_{y}^{\prime}(x_{0}\textbf{,}y_{0})+\lambda\varphi_{y}^{\prime}(x_{0}\textbf{,}y_{0})=0\textbf{;} \\ \end{aligned}$

$\begin{aligned} &\text{因为:}\varphi_{y}^{\prime}(x_{0}\textit{,}y_{0})\neq0;\\&\text{于是根据}F_{y}^{\prime}=f_{y}^{\prime}(x_{0}\textit{,}y_{0})+\lambda\varphi_{y}^{\prime}(x_{0} \text{,}y_0)=0,\\&\text{可以解得:} \lambda=-\frac{f_{y}^{\prime}(x_{0},y_{0})}{\varphi_{y}^{\prime}(x_{0},y_{0})} \\ &\text{将}\lambda=-\frac{f_{y}^{\prime}(x_{0},y_{0})}{\varphi_{y}^{\prime}(x_{0},y_{0})}\text{代入}\\&F_{x}^{\prime}=f_{x}^{\prime}(x_{0},y_{0})+\lambda\varphi_{x}^{\prime}(x_{0},y_{0})=0, \\ &\text{得:} \\ &f_{x}^{\prime}(x_{0},y_{0})-\frac{f_{y}^{\prime}(x_{0},y_{0})}{\varphi_{y}^{\prime}(x_{0},y_{0})}\varphi_{x}^{\prime}(x_{0},y_{0})=0 \\ &\text{即:}f_{x}^{\prime}(x_{0},y_{0})=\frac{f_{y}^{\prime}(x_{0},_{0})}{\varphi_{y}^{\prime}(x_{0},y_{0})}\varphi_{x}^{\prime}(x_{0},y_{0} \\ &当f_{x}^{\prime}(x_{0},y_{0})=0\text{时,有}\\&\varphi_{x}^{\prime}(x_{0},y_{0})=0\text{或者}f_{y}^{\prime}(x_{0},y_{0})=0~, \\ &当f_{x}^{\prime}(x_{0},y_{0})\neq0\text{时,有:}\\&\varphi_{x}^{\prime}(x_{0},y_{0})f_{y}^{\prime}(x_{0},y_{0})\neq0\text{,则必有} \end{aligned}$

$\begin{aligned} &\varphi_x '(x_0,y_0)\ne0且f_y'(x_0,y_0)\ne0\\& A选项为当f_x'(x_0,y_0)=0时，f_y'(x_0,y_0)可能等于0，\\& 而不是一定等于0，故A不对。\\& B选项为当f_x'(x_0,y_0)=0时，f_y'(x_0,y_0)可能不等于0，\\& 而不是一定不等于0，故B不对。\\& C选项为当f_x'(x_0,y_0)\ne0时，f_y'(x_0,y_0)必不等于0，\\& 故C不对。\\& D选项为当f_x'(x_0,y_0)\ne0时，f_y'(x_0,y_0)必不等于0，\\& 故D对。\\& 故选D。 \end{aligned}$