Absolute Extrema

Definition (Absolute Extrema) The function is said to have an absolute maximum at if for all in the domain of . Similarly, has an absolute minimum at if for all in Collectively, absolute maxima and minima are called absolute extrema.

Proposition (Absolute Extrema) A function of two variables attains both an absolute maximum and an absolute minimum on any closed bounded set where it is continuous.

Example (Absolute Extrema) Find the absolute maximum and minimum values of the given functions over the given region.

(a)   over the rectangle  



    Solution. Since is a polynomial it is continuous on the closed bounded rectangle   therefore has both absolute maximum and minimum values. We first find the critical points by solving the system   



The only critical point is and the value of there is We look at the values of on the boundary of , which consists of four line segments as shown.



On we have and for This is an increasing function of so its minimum value is and its maximum value is On we have and on This is a decreasing function of so its maximum value is and its minimum value is On we have and on By observing that we see that the minimum value of this function is and the maximum value is Finally, on we have and on with maximum value and minimum value Thus on the boundary, the minimum value of is 0 and the maximum is 9.  We compare these values with the value at the critical point and conclude that the absolute maximum value of on is and the absolute minimum value is



(b) over the rectangle



    Solution. We compute and and set and find that is the only critical point in the interior. On for . Then  



and so yielding the point On for



and so yielding the point On for



and so yielding the point On for



and so yielding the point Finally, we have (the minimum), and (the maximum).

(c) Find the hottest and coldest points on the metal plate whose temperature is given by

    Solution. Since is continuous and is closed and bounded, we know that the absolute maximum and minimum exist. We find that



and so the only critical point in the interior of is This is a saddle point because the discriminant of at is negative. The boundary of consists of four line segments and as follows. On and we have for which achieves a maximum at and a minimum at and Similarly, on and we have for which achieves its maximum at and and its minimum at We see that the hottest points are and and the coldest points are and

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Cite this as:
Absolute Extrema Of Multivariate Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/absolute-extrema-of-multivariate-functions.html