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

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

$$\begin{gathered} \mathrm{The}\mathrm{first}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {g}_x(x,y)=\frac{\partial g}{\partial \mathrm{x}}=4{\mathrm{x}}^3-16xy\mathrm{and}{g}_y(x,y)=\frac{\partial g}{\partial \mathrm{y}}=4y^3-8x^2 \\ \mathrm{Now},\mathrm{solve}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{equations}:4{\mathrm{x}}^3-16xy=0\mathrm{and}4y^3-8x^2=0,\mathrm{we}\mathrm{get}, \\ 4x(x^2-4y)=0\mathrm{and}{y}^3=2x^2 \\ \Rightarrow x=0,x^2=4y,y^3=8y \\ \Rightarrow x=0,y=0\mathrm{and}x=3.36,y=2.82 \\ \mathrm{We}\mathrm{find}\mathrm{two}\mathrm{stationary}\mathrm{point}s\mathrm{of}g,\mathrm{namely}:(0,0),(3.36,2.82) \\ \mathrm{The}\mathrm{second}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {g}_{xx}(x,y)=\frac{{\partial }^{2}g}{\partial {\mathrm{x}}^2}=12x^2-16y,g_{yy}(x,y)=\frac{{\partial }^{2}g}{\partial {\mathrm{y}}^2}=12y^2\mathrm{and}{g}_{xy}(x,y)=\frac{{\partial }^{2}g}{\partial \mathrm{x}\partial \mathrm{y}}=-16x \\ {g}_{xx}(0,0)=0,g_{yy}(0,0)=0and{g}_{xy}(0,0)=0 \\ {g}_{xx}(3.36,2.82)=90.35,g_{yy}(3.36,2.82)=95.42and{g}_{xy}(3.36,2.82)=-53.76 \\ \mathrm{The}\mathrm{determinant}\mathrm{of}\mathrm{the}\mathrm{Hessian}\mathrm{is}: \\ det\left(H_g\left(x,y\right)\right)=\left(\frac{{\partial }^{2}g}{\partial {\mathrm{x}}^2}\right)\left(\frac{{\partial }^{2}g}{\partial {\mathrm{y}}^2}\right)-{\left(\frac{{\partial }^{2}g}{\partial \mathrm{x}\partial \mathrm{y}}\right)}^2 \\ det\left(H_g\left(0,0\right)\right)=0\times 0-0=0 \\ det\left(H_g\left(128,32\right)\right)=90.35\times 95.42-{\left(-53.76\right)}^2=5731 \end{gathered}$$

$$\begin{gathered} \mathrm{If}g\mathrm{has}\mathrm{a}\mathrm{stationary}\mathrm{point}\mathrm{at}({\mathrm{x}}_0,{\mathrm{y}}_0),\mathrm{then} \\ \left(\mathrm{a}\right)g\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{maximum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_g\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {g}_{xx}(x_0,y_0)<0or{g}_{yy}(x_0,y_0)<0. \\ \left(\mathrm{b}\right)g\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{minimum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_g\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {g}_{xx}(x_0,y_0)>0\mathrm{or}{g}_{yy}(x_0,y_0)>0. \\ \left(\mathrm{c}\right)g\mathrm{has}\mathrm{a}\mathrm{saddle}\mathrm{point}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_g\right(x_0,y_0\left)\right)<0. \\ \left(\mathrm{d}\right)\mathrm{If}det\left(H_g\right(x_0,y_0\left)\right)=0,\mathrm{no}\mathrm{conclusion}\mathrm{may}\mathrm{be}\mathrm{drawn}\mathrm{about}\mathrm{the}\mathrm{behavior}\mathrm{of} \\ g\mathrm{at}(x_0,y_0). \\ \mathrm{In}\mathrm{the}\mathrm{given}\mathrm{function},det\left(H_g\left(0,0\right)\right)=0.\mathrm{Hence},\mathrm{no}\mathrm{conclusion}\mathrm{can}\mathrm{be}\mathrm{made}. \\ g\left(0,0\right)=0^4-8{\left(0\right)}^2\left(0\right)+0^4=0 \\ \mathrm{Also},det\left(H_g(3.36,2.82)\right)>0\mathrm{with}{g}_{xx}(3.36,2.82)>0\mathrm{and}{g}_{yy}(3.36,2.82)>0 \\ \mathrm{Hence},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{minimum}\mathrm{at}(3.36,2.82)\mathrm{with}\mathrm{minimum}\mathrm{value}, \\ g(3.36,2.82)={\left(3.36\right)}^4-8{\left(3.36\right)}^2\left(2.82\right)+{\left(2.82\right)}^4=-64 \end{gathered}$$

$$\begin{gathered} \mathrm{When}x=0,\underset{y\to \infty }{\lim }f(0,y)=\infty \mathrm{and}\underset{y\to -\infty }{\lim }f(0,y)=\infty \\ \mathrm{Therefore},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{absolute}\mathrm{minimum}\mathrm{at}(3.36,2.82) \end{gathered}$$