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May 18th, 2018, 06:29 AM   #1
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Post Classification of critical points

My question is to find critical points of:
xy^2 - x^2 +3x—y^4+ze^(-z)
and classify them. Please help, I have problems with finding the critical points, because the system of equation becomes big.
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May 18th, 2018, 07:23 AM   #2
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Three equations is really that big a system! You have $\displaystyle f(x, y, z)= xy^2- x^2+ 3x- 4y^4+ ze^{-z}$. Setting the partial derivatives to 0:
$\displaystyle f_x= y^2- 2x+ 3= 0$
$\displaystyle f_y= 2xy- 16y^3= 0$
$\displaystyle f_z= e^{-z}- ze^{-z}= 0$

The third equation involves only z: $\displaystyle e^{-z}(1- z)= 0$. Since $\displaystyle e^{-z}$ is never 0, z= 1.

We now have $\displaystyle y^2- 2x+ 3= 0$ and $\displaystyle 2xy- 16y^3= y(2x- 16y^2)= 0$. From the second, either y= 0 or $\displaystyle 2x=16y^2$.

If y= 0, the first equation becomes -2x+ 3= 0 so x= 3/2. x= 3/2, y= 0, z= 1 is one solution.

If $\displaystyle 2x= 16y^2$ then $\displaystyle x= 8y^2$ and the first equation becomes $\displaystyle y^2- 2(8y^2)+ 3= -15y^2+ 3= 0$. $\displaystyle 15y^2= 3$ so $\displaystyle y^2= 1/5$and y is either $\displaystyle \frac{\sqrt{5}}{5}$ or $\displaystyle -\frac{\sqrt{5}}{5}$.

In the either case $\displaystyle x= 8y^2= \frac{8}{5}$.

So $\displaystyle x= \frac{8}{5}$, $\displaystyle y= \frac{\sqrt{5}}{5}$, $\displaystyle z= 1$ is another solution and [math]$\displaystyle x= \frac{8}{5}$, $\displaystyle y= -\frac{\sqrt{5}}{5}$, $\displaystyle z= 1$ is a third solution.
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