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Unformatted text preview: 19. Max-Min Applications P. K. Lamm 10/30/11 (20:40) p. 1 / 8 Lecture Notes: 19. Max-Min Applications These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Finding Absolute Extrema of Continuous Functions In the applied problems found in this section, the goal is to determine the absolute max or absolute min of a given continuous function. The steps needed to accomplish goal this depend on whether or not the function is defined on a closed interval [ a,b ]. • Find the absolute extrema of a continuous function over a closed interval. Recall that a continuous function f defined on a closed interval [ a,b ] always attains its absolute max/min value. To find the values of x for which this occurs, the steps are: 1. Find all critical points of f in the interval [ a,b ]. 2. Evaluate f at these critical points and at the endpoints x = a , x = b , to determine where among these points f attains its absolute max and absolute min values. • Find the absolute extrema of a continuous function defined over an interval that is not closed. In this case, our standard techniques allow us to find locations of local max or min values of the function, but other arguments must be used to show that the local extremum is actually an absolute extremum. For example, if there is only one critical point c in the open interval ( a,b ) on which a given function f is defined, and if f is such that f 00 ( x ) > for all x ∈ ( a,b ), then not only is f concave up at the critical point c (ensuring that f has a local min at c ), but it is concave up on the entire interval ( a,b ). Then in this situation, the local min value of f at x = c is actually the absolute min value of f on the interval ( a,b ). Or, alternatively, a sign chart for the first derivative f of f could be used to show that f is decreasing on ( a,c ) and increasing on ( c,b ), where c is the only critical point of f in ( a,b ). Then the First Derivative Test guarantees that f has a local min at c , but the information about the graph gained from the sign chart confirms that f actually has an absolute min at x = c . The applications which follow illustrate the above techniques for functions defined on closed in- tervals such as [ a,b ] as well as for functions defined on open intervals such as ( a,b ), (-∞ , ∞ ), ( a, ∞ ), etc.....
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- Fall '10
- Calculus, P. K. Lamm, Max-Min Applications