Derivatives-Global Max-Min

[David1717]

A function w=3x^4+3x^2y-y3
It has 'saddle point' - (0,0) and point of minimum (-1/2,-1/2).
How I can check whether it is local OR global minima? (without setting a graph)

[jim360]

Does y3 mean y cubed ?

[David1717]

Yes, y^3 is y cubed
Sorry for this mistake!

[jim360]

That's all right.

In general a minimum (maximum) is global if there exists no point in the function that takes a lower (higher) value than at the particular minimum. One way to check that a minimum is only local therefore is to find a point that takes a lower value somewhere.

At (-1/2,-1/2) the value of w is 3*(-0.5^4) +3*(-0.5)^2*(-0.5) - (-0.5)^3 = -0.0625 (I think). However at the point x=0, y = 5, say, w = -5^3 = -125 which is lower. So the minimum at (-1/2,-1/2) is only local.

In general it's possible to see that a minimum or maximum is global if the graph has a form such that for large values of the variables the function contiunes to grow away from the minimum. So signs are important, for example. Here, for example, y^3 is an odd function so covers the whole range of values for w.