In Exercises 31–52, find the relative maxima, relative minima, and saddle points for the given functions. Determine whether the function has an absolute maximum or absolute minimum as well.$g(x,y)=8x^3-3x{y}^2+2y^3-4x-2$

A function,$g(x,y)=8x^3-3x{y}^2+2y^3-4x-2$

$$\begin{gathered} \mathrm{The}\mathrm{first}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {g}_x(x,y)=\frac{\partial g}{\partial \mathrm{x}}=24{\mathrm{x}}^2-3y^2-4\mathrm{and}{g}_y(x,y)=\frac{\partial g}{\partial \mathrm{y}}=-6xy+6y^2 \\ \mathrm{Now},\mathrm{solve}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{equations}:24{\mathrm{x}}^2-3y^2-4=0\mathrm{and}-6xy+6y^2=0,\mathrm{we}\mathrm{get}, \\ x=y\mathrm{and}21x^2=4 \\ \Rightarrow x=y=\pm \frac{2}{\sqrt{21}}=\pm 0.43 \\ \mathrm{We}\mathrm{find}\mathrm{one}\mathrm{stationary}\mathrm{point}\mathrm{of}g,\mathrm{namely}:(0.43,0.43) \\ \mathrm{The}\mathrm{second}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {g}_{xx}(x,y)=\frac{{\partial }^{2}g}{\partial {\mathrm{x}}^2}=48x,g_{yy}(x,y)=\frac{{\partial }^{2}g}{\partial {\mathrm{y}}^2}=12y-6x\mathrm{and}{g}_{xy}(x,y)=\frac{{\partial }^{2}g}{\partial \mathrm{x}\partial \mathrm{y}}=-6y \\ {g}_{xx}(0.43,0.43)=20.64,g_{yy}(0.43,0.43)=2.58\mathrm{and}{g}_{xy}(0.43,0.43)=-2.58 \\ {g}_{xx}(-0.43,-0.43)=-20.64,g_{yy}(-0.43,-0.43)=-2.58\mathrm{and}{g}_{xy}(-0.43,-0.43)=2.58 \\ \mathrm{The}\mathrm{determinant}\mathrm{of}\mathrm{the}\mathrm{Hessian}\mathrm{is}: \\ det\left(H_g\left(x,y\right)\right)=\left(\frac{{\partial }^{2}g}{\partial {\mathrm{x}}^2}\right)\left(\frac{{\partial }^{2}g}{\partial {\mathrm{y}}^2}\right)-{\left(\frac{{\partial }^{2}g}{\partial \mathrm{x}\partial \mathrm{y}}\right)}^2 \\ det\left(H_g(0.43,0.43)\right)=20.64\times 2.58-{\left(-2.58\right)}^2=46.6 \\ det\left(H_g(-0.43,-0.43)\right)=\left(-20.64\right)\times \left(-2.58\right)-{\left(2.58\right)}^2=46.6 \end{gathered}$$

$$\begin{gathered} \mathrm{If}g\mathrm{has}\mathrm{a}\mathrm{stationary}\mathrm{point}\mathrm{at}({\mathrm{x}}_0,{\mathrm{y}}_0),\mathrm{then} \\ \left(\mathrm{a}\right)g\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{maximum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_g\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {g}_{xx}(x_0,y_0)<0or{g}_{yy}(x_0,y_0)<0. \\ \left(\mathrm{b}\right)g\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{minimum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_g\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {g}_{xx}(x_0,y_0)>0\mathrm{or}{g}_{yy}(x_0,y_0)>0. \\ \left(\mathrm{c}\right)g\mathrm{has}\mathrm{a}\mathrm{saddle}\mathrm{point}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_g\right(x_0,y_0\left)\right)<0. \\ \left(\mathrm{d}\right)\mathrm{If}det\left(H_g\right(x_0,y_0\left)\right)=0,\mathrm{no}\mathrm{conclusion}\mathrm{may}\mathrm{be}\mathrm{drawn}\mathrm{about}\mathrm{the}\mathrm{behavior}\mathrm{of} \\ g\mathrm{at}(x_0,y_0). \\ \mathrm{In}\mathrm{the}\mathrm{given}\mathrm{function},det\left(H_g(0.43,0.43)\right)>0\mathrm{with}{g}_{xx}(0.43,0.43)>0 \\ \mathrm{and}{g}_{yy}(0.43,0.43)>0.\mathrm{Hence},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{minimum}\mathrm{at}(0.43,0.43) \\ \mathrm{with},g(0.43,0.43)=8{\left(0.43\right)}^3-3(0.43){\left(0.43\right)}^2+2{\left(0.43\right)}^3-4(0.43)-2=-3.16 \\ \mathrm{Also},det\left(H_g(-0.43,-0.43)\right)>0\mathrm{with}{g}_{xx}(-0.43,-0.43)<0 \\ \mathrm{and}{g}_{yy}(-0.43,-0.43)<0.\mathrm{Hence},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{maximum}\mathrm{at}(-0.43,-0.43) \\ \mathrm{with},g(-0.43,-0.43)=8{\left(-0.43\right)}^3-3(-0.43){\left(-0.43\right)}^2+2{\left(-0.43\right)}^3-4(-0.43)-2=-4 \end{gathered}$$

$$\begin{gathered} \mathrm{When}y=0,\underset{x\to \infty }{\lim }f(x,0)=\infty \mathrm{and}\underset{x\to -\infty }{\lim }f(x,0)=-\infty \\ \mathrm{Therefore},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{local}\mathrm{minimum}\mathrm{at}(0.43,0.43) \\ \mathrm{and}\mathrm{local}\mathrm{maximum}\mathrm{at}(-0.43,-0.43) \end{gathered}$$