91Ó°ÊÓ

A function,$f(x,y)=x^3−12xy+y^3$

$$\begin{gathered} \mathrm{The}\mathrm{first}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {f}_x(x,y)=\frac{∂\mathrm{f}}{∂\mathrm{x}}=3{\mathrm{x}}^2-12y\mathrm{and}{f}_y(x,y)=\frac{∂\mathrm{f}}{∂\mathrm{y}}=3{\mathrm{y}}^2-12x \\ \mathrm{Now},\mathrm{solve}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{equations}:3{\mathrm{x}}^2-12y=0\mathrm{and}3{\mathrm{y}}^2-12x=0,\mathrm{we}\mathrm{get}, \\ {\mathrm{x}}^2=4y\mathrm{and}{\mathrm{y}}^2=4x \\ ⇒{\left(\frac{x^2}{4}\right)}^2=4x \\ ⇒x\left(x^3-64\right)=0 \\ ⇒x=0,y=0\mathrm{and}x=4,y=4 \\ \mathrm{We}\mathrm{find}\mathrm{two}\mathrm{stationary}\mathrm{point}s\mathrm{of}f,\mathrm{namely}:(0,0),(4,4) \\ \mathrm{The}\mathrm{second}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {f}_{xx}(x,y)=\frac{{∂}^2\mathrm{f}}{∂{\mathrm{x}}^2}=6x-12,f_{yy}(x,y)=\frac{{∂}^2\mathrm{f}}{∂{\mathrm{y}}^2}=6y-12\mathrm{and}{f}_{xy}(x,y)=\frac{{∂}^2\mathrm{f}}{∂\mathrm{x}∂\mathrm{y}}=-12 \\ {f}_{xx}(0,0)=-12,f_{yy}(0,0)=-12\mathrm{and}{f}_{xy}(0,0)=-12 \\ {f}_{xx}(4,4)=12,f_{yy}(4,4)=12\mathrm{and}{f}_{xy}(4,4)=-12 \\ \mathrm{The}\mathrm{determinant}\mathrm{of}\mathrm{the}\mathrm{Hessian}\mathrm{is}: \\ det\left(H_f\left(x,y\right)\right)=\left(\frac{{∂}^2\mathrm{f}}{∂{\mathrm{x}}^2}\right)\left(\frac{{∂}^2\mathrm{f}}{∂{\mathrm{y}}^2}\right)-{\left(\frac{{∂}^2\mathrm{f}}{∂\mathrm{x}∂\mathrm{y}}\right)}^2 \\ det\left(H_f\left(0,0\right)\right)=\left(-12\right)×\left(-12\right)-{\left(-12\right)}^2=0 \\ det\left(H_f\left(4,4\right)\right)=12×12-{\left(-12\right)}^2=0 \end{gathered}$$

$$\begin{gathered} \mathrm{If}f\mathrm{has}\mathrm{a}\mathrm{stationary}\mathrm{point}\mathrm{at}({\mathrm{x}}_0,{\mathrm{y}}_0),\mathrm{then} \\ \left(\mathrm{a}\right)f\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{maximum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_f\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {f}_{xx}(x_0,y_0)<0or{f}_{yy}(x_0,y_0)<0. \\ \left(\mathrm{b}\right)f\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{minimum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_f\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {f}_{xx}(x_0,y_0)>0\mathrm{or}{f}_{yy}(x_0,y_0)>0. \\ \left(\mathrm{c}\right)f\mathrm{has}\mathrm{a}\mathrm{saddle}\mathrm{point}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_f\right(x_0,y_0\left)\right)<0. \\ \left(\mathrm{d}\right)\mathrm{If}det\left(H_f\right(x_0,y_0\left)\right)=0,\mathrm{no}\mathrm{conclusion}\mathrm{may}\mathrm{be}\mathrm{drawn}\mathrm{about}\mathrm{the}\mathrm{behavior}\mathrm{of} \\ f\mathrm{at}(x_0,y_0). \\ \mathrm{In}\mathrm{the}\mathrm{given}\mathrm{function},det\left(H_f\left(0,0\right)\right)=0,\mathrm{hence},\mathrm{no}\mathrm{conclusion}\mathrm{can}\mathrm{be}\mathrm{made} \\ \mathrm{Also},det\left(H_f\left(4,4\right)\right)=0,\mathrm{hence},\mathrm{no}\mathrm{conclusion}\mathrm{can}\mathrm{be}\mathrm{made} \end{gathered}$$

There are no maximum and minimum points.