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 Calculus Calculus Math Forum

 May 18th, 2018, 06:29 AM #1 Newbie   Joined: May 2018 From: London Posts: 1 Thanks: 0 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. May 18th, 2018, 07:23 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 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. Tags classification, critical, points Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Dan3500 Calculus 1 August 17th, 2016 04:17 PM irvm Calculus 6 January 15th, 2016 05:54 AM mathkid Calculus 1 November 11th, 2012 06:34 PM Timk Calculus 3 November 29th, 2011 10:59 AM summerset353 Calculus 1 March 5th, 2010 01:50 AM

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