Question

1. which of the following is the correct formula to be used in the second derivatives test?(single choice)*

d=(f_xy)(f_yx)-(f_xx)^2
d=(f_xy)(f_xx)-(f_yy)^2
d=(f_xy)(f_yy)-(f_xx)^2
d=(f_xx)(f_yy)-(f_xy)^2

Explanation:

In the Second - Derivatives Test for a function of two variables $z = f(x,y)$, the discriminant $D$ is defined as $D = f_{xx}f_{yy}-(f_{xy})^2$, where $f_{xx}=\frac{\partial^{2}f}{\partial x^{2}}$, $f_{yy}=\frac{\partial^{2}f}{\partial y^{2}}$ and $f_{xy}=\frac{\partial^{2}f}{\partial x\partial y}$.

Answer:

D. $D=(f_{xx})(f_{yy})-(f_{xy})^2$