Absolute Extrema

    The absolute maximum and absolute minimum of a function is defined. Examples are given of each type for some familiar functions. The extreme value theorem is stated and it is shown (through examples) how the assumptions in the extreme value theorem are necessary. Relative extremum of a function are also defined. The critical number theorem will state that a relative extrema of a continuous function can occur only at a critical number, but it does not say that a relative extrema must occur at each critical number. The classical example is the cubic function; the derivative is 0 at 0 but there is no relative extrema at 0. The cubic function merely has a horizontal tangent at 0.  In summary, given a continuous function on a closed bounded interval, to find the absolute extrema we first find all critical numbers and then compute the values of the given function at these numbers and on the boundary.

Definition (Absolute Maximum) Let be a function with domain If for all in then has an absolute maximum at Sometimes an absolute maximum is called a global maximum.

Definition (Absolute Minimum) Let be a function with domain If for all in then has an absolute minimum at Sometimes an absolute minimum is called a global minimum.

Definition (Absolute Extrema) Let be a function with domain If has either an absolute maximum or a absolute minimum at then has an absolute extrema at , and we say that is an extreme value.  

Example (Absolute Maximum and Minimum Values) State whether the following functions have absolute extrema on their domains.

(a) Linear functions  

    Solution. Linear functions do not have absolute extrema unless in the trivial case of

(b) Quadratic functions  

    Solution. Quadratic functions have an absolute maximum or absolute minimum depending on the sign of The vertex is given by and is always an absolute maximum (if ) or absolute minimum (if ).

(c) The sine and cosine functions.

    Solution. The trigonometric functions and have global maximum and global minimum; and since these functions are periodic they attain these values periodically.

Proposition (Extreme Value Theorem) If is a continuous function on a closed bounded interval then attains an absolute maximum value and an absolute minimum value at some numbers and in

    The extreme value theorem is an existence theorem because the theorem tells of the existence of maximum and minimum values but does not show how to find it.

Example (Extreme Value Theorem) State whether the function has absolute extrema on its domain:

             

    Solution. The function does not have a global maximum since it takes on all values less than, but arbitrarily close to, 0. However, it never reaches the value of This function is not continuous on a closed bounded interval containing 0.

Example (Extreme Value Theorem)  State whether the function has absolute extrema on its domain:



    Solution. The function does not have a global minimum since it takes on all values greater than, but arbitrarily close to, 0. However, it never reaches the value of This function is not continuous on a closed bounded interval containing 0.

• print this page
• pages linking here
• related pages

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