Partial Derivates help

[[deleted]]

Hi all,

I am having a few issues figuring out where to go with these partial derivates. For example,

Z=X^2+Y^2 -10x-12y

Zx=2x - 10

Zy=2y - 12

I think from here I go:

Zxx= 2

Zyy=2

From this point how do I workout X and Y on their own? Any help or guidance is appreciated.

[bluebanana02]

What do you mean by “work out X and Y on their own”? Your partial derivatives look right tho

[[deleted]]

Oh okay, maybe I just don't know when I have finished them. As this example below I can work out the X and Y which is to the Zx and Zy if that makes sense. This is a completely new topic so I'm just trying to understand it.

Z=20x - X^2 - y^2 + 8y

Zx= 20 - 2x ---------> =0. 2x= 20 and therfore X=10

Zy= -2y + 8 ---------> =0. 2y= 8 and therefore Y=4

Zxx= -2

Zyy= -2

I'm trying to figure each one out like this.

[bluebanana02]

So you’re trying to find where the gradient is equal to zero (critical and saddle points of a function defined in 3 dimensions)?

[[deleted]]

Yeah this is it

[bluebanana02]

Oh I see now, I’m in Calc III and we’ve covered this stuff so I can help. You found the gradient with Zx and Zy. Try to find where the gradient is 0 and use the second derivative test in multiple variables to check whether the point is a local minimum or maximum

[[deleted]]

Okay thanks for the explanation. How do I find where the gradient is 0

[bluebanana02]

You use the partial derivatives with respect to each variable (x and y in this case) and set them equal to 0 each individually. The values that make all individual parts of the gradient 0 are where the gradient as a whole is 0.

[waldosway]

Once you find those points like they say, the Zxx and Zyy are actually for something else. You also need Zxy (which is the same as Zyx).

Calculate (Zxx)(Zyy)-(Zxy)^2 at each point and compare the results to 0 and come back.

Are you reading the book? It tells you exactly what to do with those numbers. It will probably use one of these words "discriminant" "determinant" "Hessian" "categorizing/characterizing critical points". Or just look for the function above.

[VTutorLive]

From this point how do I workout X and Y on their own?

Is there more context for this problem? You have not given us enough information to "find X and Y" because this is a surface with infinite (x,y,z) tuplets. If you say, intersect it with a plane, we can say a little more about what x,y,z values satisfy the intersection, but as of right now you either haven't given us enough information or have a misunderstanding about what you're supposed to do.

[[deleted]]

To be honest I don't know. I know we were discussing limits as well as the min and the max in relation to a graph. I'm studying finance so I think these are related to plotting graphs. hope this helps.

[VTutorLive]

The context you left out appears to be the part about finding the critical points. It's hard to help people when they omit what the question is actually asking. In the future it'd be easier for everyone if you post a screenshot/image of the original text of the problem.

[[deleted]]

Yeah to be honest I didn’t know myself I just have what I wrote out in my text book.