Minimum and Maximum help

[Math_Struggles]

Can someone please help me find the minimum and maximum of the following equation:

 

(3x^2 + 2xy + y^2)/(x^2 - 2xy + 3y^2)

 

 

Please show work!

 

 

 

Thanks!

[Cap'n Refsmmat]

We're not going to solve the problem for you. Can you show us what you've tried, or explain where you're confused?

[Mr Skeptic]

If you know calculus, it's a simple matter of finding the derivative. What would the derivative necessarily be equal to when the function is at its minimum or maximum?

[Math_Struggles]

I've simplified it to the equations

 

0 = x(x-y)^2 and 0 = y(x-y)^2. I know that x and y can't be zero because at (0,0) the function is undefined. So are the critical points anywhere where x = y? That seems to be the place that the other quantity in the two equations would be zero.

[imatfaal]

First off - some maths teachers would immediately dock points for calling the expression in your first post an equation; without an equals it isn't an equation. Secondly - do you have in your head what sort of equation (assuming it equals zero) this is - ie is it possible to draw a simple line representing this curve on a piece of paper. Thirdly perhaps give a bit more detail about what you did to simplify.

[Cap'n Refsmmat]

I've simplified it to the equations

 

0 = x(x-y)^2 and 0 = y(x-y)^2. I know that x and y can't be zero because at (0,0) the function is undefined. So are the critical points anywhere where x = y? That seems to be the place that the other quantity in the two equations would be zero.

If those are indeed the derivatives, that makes sense. I presume you took the gradient and set its components to zero?

[Math_Struggles]

Yes that is what I did. I took the partial derivatives with respect to x and y and thensimplified them down to that result.

 

I'll attemt to show my work here, but I don't know how to make the work look like real math steps rather than typed out math.

 

f(x) = partial derivative with respect to x f(y) = same with y

 

f(X) = 0 = 2((x^2-2xy+3y^2)(3x+y) - (3x^2+2xy+y^2)(x-y))/(x^2-2xy+3y^2)^2

 

f(x) = 0 = (3x^3-6x^2y+9xy^2+x^2y-2xy^2+3y^3-3x^3-2x^2y-xy^2+3x^2y+2xy^2+y^3)/(x^2-2xy+3y^2)^2

 

f(x) = 0 = (-4x^2y+8xy^2+4y^3)/(x^2-2xy+3y^2)^2

 

f(x) = 0 = (y^3+2xy^2-x^2y)/(x^2-2xy+3y^2)^2

 

The only way that f(x) will be zero is if the numerator is zero, so we set it equal to zero:

 

0 = y^3+2xy^2-x^2y

 

0 = y(y^2+2xy-x^2)

 

I realize now that when I wrote y(x-y)^2 I was wrong in that y^2+2xy-x^2 does not factor to (x-y)^2!

 

f(y) = 0 = 2((x^2-2xy+3y^2)(3x+y) - (3x^2+2xy+y^2)(x-y))/(x^2-2xy+3y^2)^2

The work is very similar for f(y), and I end up with

 

0 = x(x^2-2xy-y^2)

 

So now I don't know where to go from there. I see that the f(x) equation can be rewritten as 0 = y(x^2-2xy-y^2), but I don't know how that really helps matters.

 

 

Thanks!

 

 

[Xittenn]

Yes that is what I did. I took the partial derivatives with respect to [math] x [/math] and [math] y [/math] and thensimplified them down to that result.

 

I'll attemt to show my work here, but I don't know how to make the work look like real math steps rather than typed out math.

 

[math] f(x) = [/math] partial derivative with respect to [math] x [/math]. [math] f(y) = [/math] same with [math] y [/math]

 

 

[math] f(x) = 0 = 2 \frac{(x^2-2xy+3y^2)(3x+y) - (3x^2+2xy+y^2)(x-y)}{(x^2-2xy+3y^2)^2} [/math]

 

 

[math] f(x) = 0 = \frac{3x^3-6x^2y+9xy^2+x^2y-2xy^2+3y^3-3x^3-2x^2y-xy^2+3x^2y+2xy^2+y^3}{(x^2-2xy+3y^2)^2} [/math]

 

 

[math] f(x) = 0 = \frac{-4x^2y+8xy^2+4y^3}{(x^2-2xy+3y^2)^2} [/math]

 

 

[math] f(x) = 0 = \frac{y^3+2xy^2-x^2y}{(x^2-2xy+3y^2)^2} [/math]

 

 

The only way that [math] f(x) [/math] will be zero is if the numerator is zero, so we set it equal to zero:

 

[math] 0 = y^3+2xy^2-x^2y [/math]

 

 

[math] 0 = y(y^2+2xy-x^2) [/math]

 

 

I realize now that when I wrote [math] y(x-y)^2 [/math] I was wrong in that [math] y^2+2xy-x^2 [/math] does not factor to [math] (x-y)^2 [/math]!

 

 

[math] f(y) = 0 = 2 \frac{(x^2-2xy+3y^2)(3x+y) - (3x^2+2xy+y^2)(x-y)}{(x^2-2xy+3y^2)^2} [/math]

 

 

The work is very similar for [math] f(y) [/math], and I end up with

 

[math] 0 = x(x^2-2xy-y^2) [/math]

 

 

So now I don't know where to go from there. I see that the [math] f(x) [/math] equation can be rewritten as [math] 0 = y(x^2-2xy-y^2) [/math], but I don't know how that really helps matters.

 

 

Thanks!

 

Hi! Latex is used to format your mathematics for our viewing purposes. You can click on any example to get further details.

 

 

This sounds like a simple D-Test or Second Derivative Test. A thorough and comprehensible explanation can be found in any introductory calculus textbook.

Edited November 20, 2011 by Xittenn

[Math_Struggles]

I need to find the critical points before I can use the second derivative test. The problem is, instead of getting a number for x and y, I get expressions of one in terms of the other, and I don't know what to do with that.

 

 

[Mr Skeptic]

The only way that f(x) will be zero is if the numerator is zero, so we set it equal to zero:

0 = y(y^2+2xy-x^2)

 

The work is very similar for f(y), and I end up with

0 = x(x^2-2xy-y^2)

 

Well that is quite fortunate in that both those have the same factor, which must be zero if you eliminate the possibility where x and y are zero. Then solving the quadratic equation gives you the other answers.

[Math_Struggles]

Yeah I used the quadratic equation and got

 

[math]

 

y = (-1+\sqrt{2})x

 

[/math]

 

[math]

 

y = -(1+\sqrt{2})x

 

[/math]

 

Now I don't know how to use that to find the critical points. Is it any x and y that fulfill that? And if it's any, meaning there are infinite critical points, how do I know which is the max or min?

Edited November 21, 2011 by Math_Struggles