Consider the function: g(x,y) = 3xe^y - x^3 - e^3y
There is exactly one critical point.
This function has a local max.
The local max is NOT the absolute max.
Why is the above the case?
How is the above different from the case of a function with two variables.
I am basically trying to understand how the local max is not necessarily the absolute max and also why this is different from a function with one variable.
10 points for the best explanation.
Thanks.
Z
There is exactly one critical point.
This function has a local max.
The local max is NOT the absolute max.
Why is the above the case?
How is the above different from the case of a function with two variables.
I am basically trying to understand how the local max is not necessarily the absolute max and also why this is different from a function with one variable.
10 points for the best explanation.
Thanks.
Z
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Okay,
The difference between local, and absolute max or min is that
a local max or min has what I call "neighbors".
In other words, points to the left and right of the max or min.
Look at this graph. It is not your function, just something I made up.
http://www.flickr.com/photos/mudriver/62…
This graph shows a local max approx at x = -1.75.
The graph shows a local min near the origin
The abs max goes to infinity
The abs min goes to negative infinity
Part two, difference between two and three variable functions.
Two variable functions are two dimensional, you only have x and y coordinates.
As you know f(x) = y.
However f(x,y) = z
Your function can be written z = 3xe^y - x^3 - e^3y
This is three dimensional (the real world). 2D graphs are flat and have no height
3D graphs can be real world objects.
The difference between local, and absolute max or min is that
a local max or min has what I call "neighbors".
In other words, points to the left and right of the max or min.
Look at this graph. It is not your function, just something I made up.
http://www.flickr.com/photos/mudriver/62…
This graph shows a local max approx at x = -1.75.
The graph shows a local min near the origin
The abs max goes to infinity
The abs min goes to negative infinity
Part two, difference between two and three variable functions.
Two variable functions are two dimensional, you only have x and y coordinates.
As you know f(x) = y.
However f(x,y) = z
Your function can be written z = 3xe^y - x^3 - e^3y
This is three dimensional (the real world). 2D graphs are flat and have no height
3D graphs can be real world objects.
-
you have to plot this equation....
or you have to bend a sheet of plastic like saddle shape
The critical point in this equation is the same as saddle point.
Local max is one point in that plane where g is maximum.
There is no maximum point in this equation
You have to check this picture then you'll understand.
http://www.victusspiritus.com/wp-content/uploads/2009/12/Saddle_point.png
The general answer for your question is in that picture (3 variable, x y and g)
or you have to bend a sheet of plastic like saddle shape
The critical point in this equation is the same as saddle point.
Local max is one point in that plane where g is maximum.
There is no maximum point in this equation
You have to check this picture then you'll understand.
http://www.victusspiritus.com/wp-content/uploads/2009/12/Saddle_point.png
The general answer for your question is in that picture (3 variable, x y and g)