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Finding extreme values for functions of two variables

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1.)f(x,y)= 2x^2 + xy^2 + 5x^2 + y^2

2.)f(x,y)= ysquareroot x - y^2 - x + 6y.

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Please see the attachment.

A function which is continuous on an open set can only have local extreme values at critical points where and .
To test the nature of those critical points, the function is used according to the rule:
If then is a saddle point.
If and , then is a local maximum.
If and , then is a local minimum.
If then the test is inconclusive.

On the boundary of the domain (if it has one), we look to reduce the function to a function of a single variable, then find the local maxima and minima of that one variable function.

To get the absolute max and min, we have to ...

Solution Summary

Finding the local extreme values of functions of a single variable entails investigating the stationary points where the derivative is zero. The existence and value of global extremes additionally requires investigation of what occurs toward the edges of the domain of definition. This approach can be generalized to functions of two variables, where finding stationary points involves simultaneously solving for stationary points of the partial derivatives and then applying a more generalized "second derivative" test to describe their character. The solution consists of two pages written in Word with equations in Mathtype illustrating the application of these ideas. Each step is explained.