# lagrange multiplier determinant

Can someone please explain why in my textbook they write lagrange multiplier like this : $$\begin{vmatrix} \frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}\\ \\ \frac{\partial g}{\partial x} &\frac{\partial g}{\partial y} \end{vmatrix}=0$$ I don't understand where did this determinant come from and why is this determinant equal to zero? I just know that the lagrange multiplier is : $$f_x(x_0,y_0)=\lambda g_x(x_0,y_0),f_y(x_0,y_0)=\lambda g_y(x_0,y_0)$$

• Which problem is the textbook trying to solve with Lagrange multipliers? Edit it into your question. (Presumably, we have a constraint for which $f-\lambda g$ depends on neither $x$ nor $y$.)
– J.G.
Jul 4, 2020 at 9:47

Note that $$f_i=\lambda g_i\implies f_xg_y=\lambda g_xg_y=g_xf_y$$, so the determinant is $$f_xg_y-g_xf_y=0$$.