If the first partial derivative of a Lagrangian function of two decision variables are equal to zero at a point, then the point is A) a saddle point maximum if the mixed partial is negative B) a local maximum if the second partials are negative and $\mathrm{D}0$ C) a global maximum if the second and mixed partials are all positive D) a global minimum if the second partials are positive and $\mathrm{D}

Question

If the first partial derivative of a Lagrangian function of two decision variables are equal to zero at a point, then the point is
A) a saddle point maximum if the mixed partial is negative
B) a local maximum if the second partials are negative and $\mathrm{D}>0$
C) a global maximum if the second and mixed partials are all positive
D) a global minimum if the second partials are positive and $\mathrm{D}<0$

Quizplus · Accepted Answer

To determine the nature of a stationary point (where the first partial derivatives are zero) in a function of two variables, we use the second partial derivatives and the determinant of the Hessian matrix ($\mathrm{D}$). A point is a local maximum if the second partial derivatives are negative and the determinant of the Hessian matrix ($\mathrm{D}$) is positive, indicating a concave down curvature in both directions at that point.

If the First Partial Derivative of a Lagrangian Function of Two