Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.
6. f(x, y) = xy − 2x − 2y – x2 – y2
To find: The local maximum, local minimum and saddle point of the
Second Derivative Test:
“Suppose the second partial derivatives of f are continuous on a disk with center , and suppose that and (that is is a critical point of f). Let
If and , then is a local minimum.
If and , then is a local maximum.
If , then is not a local maximum or minimum and it is called a saddle point”.
The given function is, .
Take the partial derivative of the given function with respect to x and obtain .
Thus, . (1)
Take the partial derivative of the given function with respect to y and obtain .
Thus, . (2)
Solve the equations (1) and (2) and find the values of x and y.
From the equation (1),
Substitute the value in the equation (2) and obtain the value ofx.
Substitute the value of in the equation and obtain y value.
Thus, the critical point is,
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