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# Partial Differentiation - Stationary Points Watch

1. Hey, I've been asked to find the stationary points of the function

f(x,y) = (1/3)x^3 + (1/3)y^3 - 4xy^2 + 15y = 0

I have found the partial derivatives.

df/dx = x^2 - 4y^2

df/dy = y^2 - 8xy + 15

I know that the stationary points occure when df/dx = df/dy = 0

how do I solve these derivates? I tried rearranging it but that didn't work.
2. (Original post by sarah_vickers)
Hey, I've been asked to find the stationary points of the function

f(x,y) = (1/3)x^3 + (1/3)y^3 - 4xy^2 + 15y = 0

I have found the partial derivatives.

df/dx = x^2 - 4y^2

df/dy = y^2 - 8xy + 15

I know that the stationary points occure when df/dx = df/dy = 0

how do I solve these derivates? I tried rearranging it but that didn't work.
From the first equation
Substitute it into the second
3. (Original post by ztibor)
From the first equation
Substitute it into the second
when substituting x = 2y into df/dy I get y = +/- 1

when substituting x = -2y into df/dy I get y = sqrt(-15/17) which cannot be defined.

so my stationary points are just

(2y,1), (2y,-1), (-2y,1) and (-2y,-1) which will give me (2,1), (-2,-1), (-2,1) and (2,-1)

is that correct?
4. (Original post by sarah_vickers)
when substituting x = 2y into df/dy I get y = +/- 1

when substituting x = -2y into df/dy I get y = sqrt(-15/17) which cannot be defined.

so my stationary points are just

(2y,1), (2y,-1), (-2y,1) and (-2y,-1) which will give me (2,1), (-2,-1), (-2,1) and (2,-1)

is that correct?
No. AS you wrote the last two are not solutions
as for both points x=-2y
and if x=-2y then y will complex value and will not 1 or -1.
5. (Original post by ztibor)
No. AS you wrote the last two are not solutions
as for both points x=-2y
and if x=-2y then y will complex value and will not 1 or -1.
so my stationary points are just (2,1) and (-2,-1)?
6. how do you find the nature of the turning points? max or min or saddle?
7. my teacher has clarified that (2,1) and (-2,-1) are the stationary points.

I have found the nature of the stationary points to both be saddlepoints. Is the nature correct? My teacher has not talked through this yet.

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Updated: January 25, 2011
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