Stationary Point Problem.

[Wynn]

I have encoutered a problem that I am unable to solve.
We have a curve plotted on a graph where z is a function of x and y z=f(x,y)
And the equation of the graph looks something like this:
z = Ay²+By³-Cy⁴-x2
​(Where A B and C are constants)
I am asked to find the co-ordinates for all stationary points on the graph (we're not worrying about the nature of these points)

So I take the partial derivatives of z wrt y and x, set them equal to zero, and obtain:
∂z/∂x =-2x = 0
∂z/∂y = 2Ay +3By² - 4Cy³ = 0

But now I don't know how to continue to solve for y. x is clearly 0, but I don't know how to deal with y.
What am I doing wrong or not doing here?

 

[Ishuda]

I have encoutered a problem that I am unable to solve.
We have a curve plotted on a graph where z is a function of x and y z=f(x,y)
And the equation of the graph looks something like this:
z = Ay²+By³-Cy⁴-x2
​(Where A B and C are constants)
I am asked to find the co-ordinates for all stationary points on the graph (we're not worrying about the nature of these points)

So I take the partial derivatives of z wrt y and x, set them equal to zero, and obtain:
∂z/∂x =-2x = 0
∂z/∂y = 2Ay +3By² - 4Cy³ = 0

But now I don't know how to continue to solve for y. x is clearly 0, but I don't know how to deal with y.
What am I doing wrong or not doing here?

What are the roots of
2Ay +3By² - 4Cy³ = 0
First, factor the equation
y ( 2A +3By - 4Cy² ) = 0
Now you have either y=0 or
2A +3By - 4Cy² = 0
whose two solutions can be obtained by the quadratic formula if C is not zero [you might also need to consider cases where A, B, and/or C are zero]. Call those two solutions y₁ and y₂. So you have three places where both ∂z/∂x and ∂z/∂y are zero at the same time. What are they?

 

[Wynn]

What are the roots of
2Ay +3By² - 4Cy³ = 0
First, factor the equation
y ( 2A +3By - 4Cy² ) = 0
Now you have either y=0 or
2A +3By - 4Cy² = 0
whose two solutions can be obtained by the quadratic formula if C is not zero [you might also need to consider cases where A, B, and/or C are zero]. Call those two solutions y₁ and y₂. So you have three places where both ∂z/∂x and ∂z/∂y are zero at the same time. What are they?

the (x,y) co-ordinates of the three roots would be:
(0,0), (0,y₁) and (0,y₂)
right?
My brains was getting confused when it came to finding y as the quadratic formula only applied to the equation in parenthesis in the factored equation:
y ( 2A +3By - 4Cy² ) = 0
I thought you still had to do something with the y coefficient to find the true value of y but now that I type it out, that wouldn't make much sense.
Thanks!

 

[Ishuda]

the (x,y) co-ordinates of the three roots would be:
(0,0), (0,y₁) and (0,y₂)
right?
My brains was getting confused when it came to finding y as the quadratic formula only applied to the equation in parenthesis in the factored equation:
y ( 2A +3By - 4Cy² ) = 0
I thought you still had to do something with the y coefficient to find the true value of y but now that I type it out, that wouldn't make much sense.
Thanks!

If C is not zero, there are two roots and (0,0), (0,y₁) and (0,y₂) are the stationary points. In certain conditions y₁ and y₂ could be the same value and, depending on other circumstances, only one root might exist or one or both could be complex.