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Determine if [itex]f(x,y) = x^2+y^2[/itex] has a maximum and a minimum when we have the constraint [itex]2x^3+3x^{2}y+3xy^{2}+2y^3=1[/itex]. (1)

ATTEMPT TO SOLUTION:

A standard way of solving these kinds of problems is by using the Lagrangian multiplier-method. It consists of comparing the gradient of the function with the gradient of the constraint. However, sometimes, as in the case of [itex][/itex] with the constraint [itex][/itex], one can rapidly observe that it lacks a maximum value. This is the case of the function [itex]f(x,y) = x^{2}y[/itex] with the constraint [itex]x+y=1[/itex]. As x approaches infinity, on that line x+y=1, the function approaches infinity. Is there any similar way to observe problem (1)? How do I prove that it has or hasn't a maximum?

I've used the Lagrangian multiplier method, and I get the solution [itex]x=\frac{1}{10^{1/3}} = |y|[/itex]. I have not examined that it is correct, and I have not checked if it is a max or min.

How do I prove that this function has or has not a max or min?

[EDIT1:]

Okay. So we can interpret the question geometrically as: what is the smallest or largest distance from the origin to that curve? And: does there exist a smallest or largest distance from the origin to that curve? Hm... I can't visualize curve (1). If it is "closed", like an ellipse or circle, then there certainly has to exist a smallest and largest distance. But if it is like a line - which extends itself infinitely - there is a smallest distance, but not a largest. So the question is: how do I know the expression (1) is for a curve of finite length?

[EDIT2:]

Okay guise. The following inequalities are true:

[itex](x^3-1)^2 = x^6+1-2x^3 \geq 0[/itex]

[itex]\Rightarrow x^6+1 \geq 2x^3[/itex]

[itex](x^2-y) = x^4+y^2-2x^{2}y \geq 0[/itex]

[itex]\Rightarrow \frac{3}{2} (x^4+y^2)\geq 3x^{2}y[/itex]

Add a little symmetry-reasoning, and we get an inequality which can be used to study the constraint (1). We get:

[itex]2+x^6+y^6+\frac{3}{2}(x^4+y^4+x^2+y^2) \geq 2(x^3+y^3)+3(x^{2}y+xy^{2})=1[/itex]

Where the right side is the constraint (1), written alittle more nicely. So... what conclusions can I make out of this? If I subtract 1 from both sides of the inequality, I get an expression on the left which always is positive, independent of choice of (x,y) and on the right I just get something which is equal to zero. Is this enough to prove that there is no bound for the set of points (1), and that therefore, it is not compact (the curve is not of finite length)?

I find it hard to accept this kind of reasoning, since Ido not clearly provethat there is no bound for the set of points (1). But I don't know guise. I would appreciate it if somebody could help me interpret this stuff, and maybe find a formal way of proving that that set of points is not compact...

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# Homework Help: Optimization problem with constraint

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