 CMIIh

2020-12-24

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.
$f\left(x,y\right)=y{e}^{x}-{e}^{y}$ 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,
$f\left(0,0\right)=\left(0\right){e}^{0}-{e}^{0}=-{e}^{0}=-1$