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Derivatives-Global Max-Min
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)
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)
Answers
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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) +
15:30 Tue 15th Oct 2013
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.
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.
1 to 3 of 3
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