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11. 7 Maximum and Minimum Values
The goal of this section is to find the maxima and minima of a function of two variables. RECALL: In 2 D (last year) critical points are found where derivatives equal zero or don't exist. Local Maxima: A function of two variables has a local maximum at ( a, b ) if f ( x, y ) ≤ f ( a, b ) when ( x, y ) is near ( a, b ). Another way to look at it - We say that f ( x, y ) has a local maximum at a point ( p, q ) if there is a circle centered at ( p, q ) such that f ( x, y ) ≤ f ( p, q ) for all ( x, y ) in the circle.
Local Minima: A function of two variables has a local minimum at ( a, b ) if f ( x, y ) ≥ f ( a, b ) when ( x, y ) is near ( a, b ). Local Extrema: To find the local extrema of a function of two variables which is smooth everywhere, we first notice that the tangent plane to the graph of f ( x, y ) is horizontal at a local maximum or local minimum.
Some critical points yield saddle points, which are NEITHER relative maxima nor relative minima. Some 3 D functions require a more sophisticated test to determine whether a critical point yields a maxima, minima, or neither. This method of classifying critical points uses the determinant of the second partials
Second Partials Test
If ( xo, yo ) is a critical point of a function f ( x, y ) whose second derivatives exist at ( xo, yo ), then a) If b) If c) If d) If D = 0 , no information has been obtained
If it rains on the surface whose height is
Where will the puddle begin to form?
EXAMPLE: Find the absolute maximum and absolute minimum values of f ( x, y ) = 5 - 2 x^2 + 2 xy - y^2 on the region R , whose coordinates satisfy 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2 EXAMPLE: Find the maximum and minimum value of f ( x, y ) = 4 x^2 - 3 y^2 + 2 xy in the circle x^2 + y^2 = 1.
