设 $f(x, y)$ 与 $\varphi(x, y)$ 均为可微函数, 且 $\varphi_y^{\prime}(x, y) \neq 0$, 已知 $\left(x_0, y_0\right)$ 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 $f_x^{\prime}\left(x_0, y_0\right)=0$, 则 $f_y^{\prime}\left(x_0, y_0\right)=0$.
B. 若 $f_x^{\prime}\left(x_0, y_0\right)=0$, 则 $f_y^{\prime}\left(x_0, y_0\right) \neq 0$.
C. 若 $f_x^{\prime}\left(x_0, y_0\right) \neq 0$, 则 $f_y^{\prime}\left(x_0, y_0\right)=0$.
D. 若 $f_x^{\prime}\left(x_0, y_0\right) \neq 0$, 则 $f_y^{\prime}\left(x_0, y_0\right) \neq 0$.