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Instructions on how to find the absolute maximum and minimum values (absolute extrema) of a continuous function in both closed and open intervals. It includes examples and exercises to help understand the concepts. The document also explains the difference between absolute and relative extrema, and provides figures to illustrate the concepts.
Typology: Lecture notes
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Absolute Maximums or Absolute Minimums (Absolute Extrema) in a Closed Interval: Let f be a continuous function on a closed interval [ a , b ].. Let c be a number in that interval. Then…
(In other words, every other y or function value is lower or less than f c ( ).)
(In other words, every other y or function value is larger or greater than f c ( ).) Note: Sometimes the textbook refers to an absolute maximum or absolute minimum as an absolute extremum.
Also of note: Just as a relative maximum or a relative minimum was the y -value or function value, so it is with absolute extrema. The x -value or the ordered pair ( x, y) is the location of an absolute or relative maximum/minimum.
Look at Figure 2 below. (This is the same figure 2 that is on page 305 of the 2nd^ half (calculus part)of your textbook.)
In picture (a), there is an absolute maximum at x 1 an absolute minimum at x 2. f ( x 1 ) is a greater function value
than all other function values ( y -values) in the closed interval [ x 1 (^) , x 3 ]. In picture (b), there is an absolute
minimum at x 1 and an absolute maximum at x 2 in the closed interval, [ x 1 (^) , x 2 ]. In picture (c), for the closed
interval [ x 1 (^) , x 5 ], the absolute minimum is f ( x 3 (^) ) at x 3 and the absolute maximum is f ( x 4 (^) ) at x 4.
Absolue Maximums or Absolute Minimums (Absolute Extrema) in an Open Interval: If a function is continuous on an open interval , there may or may not be an absolute maximum or an absolute minimum. The ideas ∞ or - ∞ cannot be absolute extrema, only exact real numbers can be absolute extrema. Examine figure 3(a) below. There is an absolute minimum at x 1 , but there is no absolute maximum value, since the greatest function value goes toward infinity. Also, a function that has a ‘break’ or ‘gap’ at a value of x may or may not have an absolute minimum or maximum. Look at figure 3(b) below. Since the function is not continuous, it may be difficult to determine absolue extrema. In the interval [ a , b ], there is an absolute minimum at x = a, but there is no absolute maximum value. The actual function value at x 1 is lower than the open circle above. There is not an absolute maximum at x 1.
We will begin with finding absolute extrema on a CLOSED INTERVAL.
To find absolute extrema of a function f on a continuous closed interval [ a, b ], follow these steps.
Ex. 1: Find the absolute minimum and absolute maximum values of f x ( ) x^2 8 x 10 on the interval [0,7].
Where do these values occur?
Ex 4: Find the absolute extrema of f ( ) x ( x 1)2 3on [2, 9] and where they occur.
Ex 5: A retailer has determined the cost C for ordering and storing x units of a product to be modeled by the cost
function,
C x ( ) 3 x x
, 1 x 200. (The delivery truck can bring at most 200 units per order.) Find the
size of the order that will minimize the cost.
Now, finding absolute extrema on an OPEN INTERVAL.
To find any possible absolute extrema on an open interval, follow these steps.
Find the absolute extrema, if they exist, as well as all ordered pairs where they occur.
Ex 6: ( ) (^2) 1
x f x x
Ex 7: g x ( ) x ln x
Ex 10:
The number of salmon swimming upstream is approximated by S x ( ) x^3^ 3 x^2 360 x 5000, 6 x 20
where x represents the temperature of the water in degrees Celsius. Find the water temperature that produces the maximum number of salmon swimming upstream.