How to find critical points for a differential equation?

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Homework Help Overview

The discussion revolves around finding critical points for the function f(x,y) = 3x^5y^2 - 30x^3y^2 + 60xy^2 + 150, specifically where this function equals zero. The subject area pertains to differential equations and multivariable calculus.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the necessity of setting the partial derivatives equal to zero to find critical points. There is also a suggestion to consider whether critical points and points where the function equals zero occur simultaneously. Some participants explore the implications of the derived equations and the relationships between the variables.

Discussion Status

The discussion is ongoing, with participants providing various approaches and interpretations. Some guidance has been offered regarding the algebraic manipulation of the derivatives, and there is an exploration of alternative methods to find critical points. Multiple interpretations of the problem are being considered.

Contextual Notes

There is mention of potential confusion regarding the relationship between critical points and the function equaling zero. Participants are also addressing possible errors in the derived equations and the need for careful consideration of the algebra involved.

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Homework Statement



f(x,y)= 3x5y2 - 30x3y2 + 60xy2 +150

How do I find the critical points where this equation equals zero?

Homework Equations



None?

The Attempt at a Solution



I found the partial derivatives, but I'm stuck now and don't know where to go from here.

dx = 15x4y2 - 90x2y2 + 60y2
dy = 6x5 - 60x3y + 10xy + 240
 
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start by setting both equal to 0! then try quadratic factoring for the dx. the dy should be obvious.
 
also, you ask for critical points and where the equation equals zero...are you sure both occur simultaneously? i haven't checked but don't assume this to be the case
 
Okay so I did that. I got for dx, x = 3 +- sqrt(20)/2
The first part of dy should actually be 6yx^5.

After I have the dx value, I can sub them into the dy equation which gives approximately y = -0.1545 and y = -3.6099. Where do I go from here? There should be 4 separate (x,y) points.
 
Spell things out a little more carefully. What you quote a solution for I think is really saying

∂f/∂x = 0 at all points along (x/y)2 = those numbers, which is actually four lines through the origin I think.

Maybe there is a nice algebraic breakup of your ∂f/∂y but anyway what you are doing should lead you to the four points along the four lines.

Alternative approach is to find (dy/dx)f and its reciprocal and find the points where both are 0. Well same thing really, -(dy/dx)f is the ratio of your two derived polynomials.

Also I think the second of yours, what you call "dy" is mistaken and it works out fairly simply.

You might as well have taken the factor 3 out of your starting equation.
 
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