Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds

Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.
$f\left(x,y\right)=y{e}^{x}-{e}^{y}$
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joshyoung05M

To find the critical point, find ${f}_{x},{f}_{y}$ and set ${f}_{x}=0,{f}_{y}=0$
${f}_{x}=t{e}^{x}=0⇒y=0$ or ${e}^{x}=0$
For ${e}^{x}=0$ have no solution for $x\in R$
Therefore y=0
Set ${f}_{y}=0$
${f}_{y}={e}^{x}-{e}^{y}=0$
Substitute $y=0,{f}_{y}={e}^{x}-{e}^{0}=0⇒{e}^{x}-1=0⇒{e}^{x}=1$
Taking ln on both sides, $\mathrm{ln}\left({e}^{x}\right)=\mathrm{ln}\left(1\right)$ , x=0
Thus (0,0) is the ritical point.
$D\left(x,y\right)={f}_{×}\left(x,y\right){f}_{yy}\left(x,y\right)-{\left({f}_{xy}\left(x,y\right)\right)}^{2}$
1) If D(a,b)>0 and ${f}_{×}\left(a,b\right)>0$ then (a,b) is local aximum of f
2) If D(a,b)>0 and ${f}_{×}\left(a,b\right)<0$ then (a,b) is local aximum of f
3) If D(a,b)<0 then f(a,b) is saddle point
4) If D(a,b)=0 then this is incobclusive
Here, ${f}_{×}=y{e}^{x},{f}_{yy}=-{e}^{y}$ and ${f}_{xy}={e}^{x}$
$D\left(x,y\right)=t{e}^{x}\left(-{e}^{y}\right)-{\left({e}^{x}\right)}^{2}=-e{y}^{x+y}-{e}^{2x}$
The critical point is(0,0)
$D\left(0,0\right)=-0{e}^{0}-{e}^{2\cdot 0}=-{e}^{0}=-1$
Thus D(0,0)=-1<0 then f(0,0) is saddle point,
$f\left(0,0\right)=\left(0\right){e}^{0}-{e}^{0}=-{e}^{0}=-1$