- #1

- 18

- 0

## Homework Statement

a. Use Implicit differentiation to find the equations of the horizontal tangent lines to the parametric equation: yx^3-2x^2y^2+xy^3=192.

b. Use Implicit differentiation to find the equations of the horizontal tangent lines to the parametric equation: yx^3-2x^2y^2+xy^3=192.

## Homework Equations

Differentiation and I used the open-source program Graph and believe that the equations of the horizontal tangent lines should be y=6 and y=-6. This is just a good approximation.

## The Attempt at a Solution

Ok, since my work spanned about 5 pages of paper in my best attempt, I will merely list the sequence of steps and my answer for each sequence.

I used implicit differentiation to determine:

dy/dx = (4xy^2-3x^2y+y^3)/(x^3-4x^2y+3xy^2)

I set the derivative equal to zero since that is the slope that my 2 tangent lines will have.

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

However, I believe I only need to set the numerator equal to 0, so:

4xy^2-3x^2y+y^3 = 0

I think this needs to be solved using some clever substitution (like the last problem I posted) and so I tried to eliminate all y-variables by expressing them in terms of x, so I can then solve for x.

From here I determine y^3=3x^2y-4xy^2

and y^2=3x^2-4xy

I used these values to substitute into the original equation and start over by taking the implicit derivative of the original equation, but now with my substitutions in place. So, now for dy/dx, I get:

dy/dx=(72x-2xy-99y)/(28x)

Again I set this equal to 0 (the slope of my tangent lines).

(72x-2xy-99y)/(28x)=0

Solving for y:

y=(72x)/(2x+99)

I now use this value of y to substitute into the original equation along with my values of y^3 and y^2. I attempt to solve for x and get to here:

(2016x^4)/(2x+99)-18x^4=192

I am stuck with the algebra here and I don't have any confidence in the answer this provides. Do I need to do the differentiation again instead of just solving for x?

I did try to differentiate and got: (1287/4)=x^2+57x

But this didn't seem to help either....

I feel like I am on the right track and probably just screwed up my arithmetic. Is my approach the right one? If I screwed up the math somewhere, I would appreciate it if someone could just point it out. If I am correct in my approach, when I go to solve the equations of the vertical tangent lines, I just need to set the denominator to 0 and do it that way, right?