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.$g(x,y)=x^2+\sin y-3$

A function,$g(x,y)=x^2+\sin y-3$

$$\begin{gathered} \mathrm{The}\mathrm{first}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {g}_x(x,y)=\frac{\partial g}{\partial \mathrm{x}}=2x\mathrm{and}{g}_y(x,y)=\frac{\partial g}{\partial \mathrm{y}}=\cos y \\ \mathrm{Now},\mathrm{solve}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{equations}:2x=0\mathrm{and}\cos y=0,\mathrm{we}\mathrm{get}, \\ x=0\mathrm{and}y=\left(2n+1\right)\frac{\mathrm{\pi }}{2} \\ \mathrm{where},n\mathrm{is}\mathrm{an}\mathrm{integer} \\ S\mathrm{tationary}\mathrm{point}s\mathrm{of}gare:\left(0,\left(2n+1\right)\frac{\mathrm{\pi }}{2}\right) \\ \mathrm{The}\mathrm{second}-\mathrm{order}\mathrm{partial}\mathrm{derivatives}\mathrm{of}\mathrm{the}\mathrm{function}\mathrm{are}: \\ {g}_{xx}(x,y)=\frac{{\partial }^{2}g}{\partial {\mathrm{x}}^2}=2,g_{yy}(x,y)=\frac{{\partial }^{2}g}{\partial {\mathrm{y}}^2}=-\sin y\mathrm{and}{g}_{xy}(x,y)=\frac{{\partial }^{2}g}{\partial \mathrm{x}\partial \mathrm{y}}=0 \\ {g}_{xx}\left(0,\left(2n+1\right)\frac{\mathrm{\pi }}{2}\right)=2,g_{yy}\left(0,\left(2n+1\right)\frac{\mathrm{\pi }}{2}\right)=\pm 1\mathrm{and}{g}_{xy}\left(0,\left(2n+1\right)\frac{\mathrm{\pi }}{2}\right)=0 \\ \mathrm{The}\mathrm{determinant}\mathrm{of}\mathrm{the}\mathrm{Hessian}\mathrm{is}: \\ det\left(H_g\left(x,y\right)\right)=\left(\frac{{\partial }^{2}g}{\partial {\mathrm{x}}^2}\right)\left(\frac{{\partial }^{2}g}{\partial {\mathrm{y}}^2}\right)-{\left(\frac{{\partial }^{2}g}{\partial \mathrm{x}\partial \mathrm{y}}\right)}^2 \\ det\left(H_g\left(0,\left(2n+1\right)\frac{\mathrm{\pi }}{2}\right)\right)=2\times \pm 1-0=\pm 2 \end{gathered}$$

$$\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,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)\right)>0\mathrm{with}{g}_{yy}\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)<0 \\ \mathrm{for}n=\mathrm{odd}\mathrm{integer} \\ \mathrm{Hence},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{minimum}\mathrm{at}\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)\mathrm{with}\mathrm{minimum}\mathrm{value}, \\ g\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)=2\left(0\right)+\sin \left(\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)-3=-4 \\ \mathrm{Also},det\left(H_g\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)\right)<0\mathrm{for}n=\mathrm{even}\mathrm{integer} \\ \mathrm{Hence},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{saddle}\mathrm{points}\mathrm{at}\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)\mathrm{with}\mathrm{value}, \\ g\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)=2\left(0\right)+\sin \left(\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right)-3=-2 \end{gathered}$$

$$\begin{gathered} \mathrm{When}y=0,\underset{x\to \infty }{\lim }f(x,0)=\infty \mathrm{and}\underset{y\to -\infty }{\lim }f(x,0)=\infty \\ \mathrm{Therefore},\mathrm{the}\mathrm{given}\mathrm{function}\mathrm{has}\mathrm{local}\mathrm{minim}a\mathrm{at}\left(0,\frac{\left(2n+1\right)\mathrm{\pi }}{2}\right) \\ \mathrm{when},n=\mathrm{odd}\mathrm{integer} \end{gathered}$$