# Finding the max points of a surface, maximize

This should not be hard at all, in fact no one seems so be having my issue so it should be me missing out on a point of information or just experience.

My mission is to with Mathematica find the x,y,z's of the maximum points of the following function

$$\left[\frac{-\left(1-x^2\right) \left(y^2-4\right)-x^2-y^2+5}{\left(x^2+y^2+1\right)^2}\right]$$

Currently I am trying with Maximize

$Maximize[\left[\frac{-\left(1-x^2\right) \left(y^2-4\right)-x^2-y^2+5}{\left(x^2+y^2+1\right)^2}\right], {x,y,z}]$

I can't make heads or tails of this answer though, so I am doing something wrong.., finding the points for reference I did with wolfram.

Wolfram Alpha gives me the points

$$\left\{\pm\frac{1}{\sqrt{3}},0,\frac{9}{8}\right\}$$

How do I use for example maximize[] or some other simple function in Mathematica to find those values? I have set the partial derivatives to 0 and solved for the points, that works but seems to me to be overly complicated.

The documentation has me confused, it does this and gets three values, I try to do the same and I get something strange

Manual:

My answer, which is to my eyes wrong:

Even if I try to use //NN I get strange answers:

I kind of do get the x coordinate if I do this but why this is the case I don't know.

$Maximize[\left[\frac{-\left(1-x^2\right) \left(y^2-4\right)-x^2-y^2+5}{\left(x^2+y^2+1\right)^2}\right], {y,z}]$

I have also tried to use FindMaximum and FindMaxValue.

This baffles me, any ideas, it should be simple right? Why don't I get the points? is it because there are two extreme points perhaps?

• People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful – Michael E2 Jun 13 '16 at 11:36
• Ok, cool, thanks Michael – Celebrin Jun 13 '16 at 14:03

I think it's only a question of accolades

Maximize[(-(1 - x^2) (y^2 - 4) - x^2 - y^2 + 5)/(x^2 + y^2 + 1)^2, {x,y}]


return

{9, {x -> 0, y -> 0}}

• Hum, accolades?, "an award or an expression of praise" merriam webster. I do know the points already through wolfram, but I cant transfer that to mathematica since I dont have wolfPRO $\left\{+-\frac{1}{\sqrt{3}},0,\frac{9}{8}\right\}$ should be the solutions, what am I not getting from your answer? – Celebrin Jun 13 '16 at 10:29
• @Celebrin Why not get the points through Mathematica instead of W|A? – Michael E2 Jun 13 '16 at 11:39
• Well that is what I hoped to do, and the question asked, . I have since calculated the points through partial derivatives.. but there should be an easier way. you got any idea Michael? – Celebrin Jun 13 '16 at 12:02
• @Celebrin. I think cyrille means "brackets". You need to remove the internal square brackets and specify only the independent variables, x and y. – m_goldberg Jun 13 '16 at 13:02
• In fact I think there is nothing wrong with Mathematica. Plot the function with PlotRange->{-1, 9.5}. You will find an absolute maximum. But there are also a number of local maxima and minima even some imaginary according to the solution by solving the system of derivatives. – cyrille.piatecki Jun 13 '16 at 13:42