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.$f(x,y)=3x^2+3x+6y^2-7$

A function,$f(x,y)=3x^2+3x+6y^2-7$

$$\begin{gathered} \mathrm{The}\mathrm{first}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {f}_x(x,y)=\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=6\mathrm{x}+3\mathrm{and}{f}_y(x,y)=\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=12\mathrm{y} \\ \mathrm{Now},\mathrm{solve}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{equations}:6\mathrm{x}+3=0\mathrm{and}12\mathrm{y}=0,\mathrm{we}\mathrm{get}, \\ \mathrm{x}=-\frac{1}{2}\mathrm{and}\mathrm{y}=0 \\ \mathrm{We}\mathrm{find}\mathrm{the}\mathrm{only}\mathrm{stationary}\mathrm{point}\mathrm{of}f,\mathrm{namely},(-\frac{1}{2},0) \\ \mathrm{The}\mathrm{second}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {f}_{xx}(x,y)=\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{x}}^2}=6,f_{yy}(x,y)=\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{y}}^2}=12\mathrm{and}{f}_{xy}(x,y)=\frac{{\partial }^2\mathrm{f}}{\partial \mathrm{x}\partial \mathrm{y}}=0 \\ \mathrm{The}\mathrm{determinant}\mathrm{of}\mathrm{the}\mathrm{Hessian}\mathrm{is}: \\ det\left(H_f\left(-\frac{1}{2},0\right)\right)=\left(\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{x}}^2}\right)\left(\frac{{\partial }^2\mathrm{f}}{\partial {\mathrm{y}}^2}\right)-{\left(\frac{{\partial }^2\mathrm{f}}{\partial \mathrm{x}\partial \mathrm{y}}\right)}^2 \\ =6\times 12-0 \\ =72 \end{gathered}$$

$$\begin{gathered} \mathrm{If}f\mathrm{has}\mathrm{a}\mathrm{stationary}\mathrm{point}\mathrm{at}({\mathrm{x}}_0,{\mathrm{y}}_0),\mathrm{then} \\ \left(\mathrm{a}\right)f\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{maximum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_f\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {f}_{xx}(x_0,y_0)<0or{f}_{yy}(x_0,y_0)<0. \\ \left(\mathrm{b}\right)f\mathrm{has}\mathrm{a}\mathrm{relative}\mathrm{minimum}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_f\right(x_0,y_0\left)\right)>0\mathrm{with} \\ {f}_{xx}(x_0,y_0)>0\mathrm{or}{f}_{yy}(x_0,y_0)>0. \\ \left(\mathrm{c}\right)f\mathrm{has}\mathrm{a}\mathrm{saddle}\mathrm{point}\mathrm{at}(x_0,y_0)\mathrm{if}det\left(H_f\right(x_0,y_0\left)\right)<0. \\ \left(\mathrm{d}\right)\mathrm{If}det\left(H_f\right(x_0,y_0\left)\right)=0,\mathrm{no}\mathrm{conclusion}\mathrm{may}\mathrm{be}\mathrm{drawn}\mathrm{about}\mathrm{the}\mathrm{behavior}\mathrm{of} \\ f\mathrm{at}(x_0,y_0). \\ \mathrm{In}\mathrm{the}\mathrm{given}\mathrm{function},det\left(H_f\left(-\frac{1}{2},0\right)\right)=72>0\mathrm{with}{f}_{xx}\left(-\frac{1}{2},0\right)=6>0\mathrm{and}{f}_{yy}\left(-\frac{1}{2},0\right)=12>0. \\ \mathrm{Hence},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{minimum}\mathrm{at}\left(-\frac{1}{2},0\right)\mathrm{with}\mathrm{minimum}\mathrm{value}, \\ f\left(-\frac{1}{2},0\right)=3{\left(-\frac{1}{2}\right)}^2+3\left(-\frac{1}{2}\right)+6{\left(0\right)}^2-7 \\ =\frac{3}{4}-\frac{3}{2}+0-7 \\ =-\frac{3}{4}-7 \\ =-\frac{31}{4} \end{gathered}$$

$$\begin{gathered} \mathrm{When}x=0,\underset{y\to \infty }{\lim }f(0,y)=\infty \mathrm{and}\underset{y\to -\infty }{\lim }f(0,y)=\infty \\ \mathrm{Therefore},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{absolute}\mathrm{minimum}\mathrm{at}\left(-\frac{1}{2},0\right) \end{gathered}$$