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Calculus/Finding the extremepoints

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Expert: - 9/6/2009


Question
Hi, could you please help me a little with this question?

Find and classify the extremepoints of this function:
f(x,y) = e^(yx^2 - 2x^2 - y^2)

I have derived it and got the following answer 0,0 -2,2 and 2,2

But I don't understand how to classify this extremepoint 0,0. How do you know whether this is a global or local extremepoint?

Thanks a lot!

Answer
The way to find out is to put the value into the second partial derivatives of x and of y.

Lets call the second parital derivatives §²f/§x² and §²f/§y².


I found that §²f/§x² = f(x,y)(y-2)(4x²(y-2)+2) and
§²f/§y² = f(x,y)(-2 + x^4 - 4x²y + 4y²).

The points to check out are (0,0), (-2,2), and (2,2).

At (0,0), §²f/§x² is negative.  This means it is a maximum in the x direction.
At (-2,2) and (2,2), §²f/§x² is 0.  This means it is an inflection point in the x direction.


I have found the §²f/§y² = f(x,y)(-2 + x^4 - 4xy + 4y²).

At (0,0), (-2,2), and (2,2), §²f/§y² is negative.
This means that all of the points are a maximum in the y direction.

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