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MA 15910 Lesson 29 Notes
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…
(a) f c ( ) is the absolute maximum of f on that interval if f x ( ) f c ( ) for every x in the interval.
(In other words, every other y or function value is lower or less than f c ( ).)
(b) f c ( ) is the absolute minimum of f on that interval if f x ( ) f c ( ) for every x in the interval.
(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.
Important: Although a function can have only one absolute minimum value
and only one absolute maximum value (in a specified closed interval), it can
have more than one location ( x values) or points (ordered pairs) where these
values occur. For any function in a closed interval [ a , b ], there will be an
absolute maximum and an absolute minimum on that interval.
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.
- Find all critical values of f in the open interval ( a, b ).
- Evaluate function f (find function values) for those critical values in ( a , b ) and the endpoints a and b of the closed interval [ a , b ].
- The largest value found is the absolute maximum for f on [ a , b }, and the smallest value found is the absolute minimum for f on [ a , b ].
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 all critical values in the open interval. Evaluate the function values at these critical values.
- Find the limits as the endpoints are approached (or limits at infinity and negative infinity). If a limit is infinity or negative infinity, these cannot be considered as the absolute extrema values.
- The greatest function value is the absolute maximum value and the least is the absolute minimum value.
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.
