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Math 1314 Lesson 13 Absolute Extrema
In earlier sections, you learned how to find relative (local) extrema. These points were the high points and low points relative to the other points around them. In this section, you will learn how to find absolute extrema, that is the highest high and/or the lowest low on the domain of the function, or on a specific closed interval.
Absolute Extrema on the Domain off
Definition: If f (x)โค f(c)for allx in the domain off, thenf(c) is
called the absolute maximum value off. If f (x)โฅ f(c)for allx
in the domain off, thenf(c) is called the absolute minimum value off.
Sometimes you will be asked to find the absolute extrema
over the interval ( โ โ, โ).
Example 1: State the absolute maximum and/or absolute minimum value(s).
Example 2: State the absolute maximum and/or absolute minimum value(s).
As you can see from these two examples, the absolute extrema may or may not exist. To find absolute extrema on
( โ โ, โ) (or on the entire domain of the given function)
algebraically, you must graph the function using the guide to curve sketching.
Absolute Extrema on a Closed Interval
More often you will be asked to find the absolute extrema over a closed interval [a, b], and, for a continuous function, those will always exist and they are much easier to find.
Theorem: If a functionf is continuous on a closed interval [a, b], thenf has both an absolute maximum value and an absolute minimum value on [a, b].
Example 5: Find the absolute maximum value and the
absolute minimum value of the function f (x)= x^2 + 2 xover the
interval [-3, 1].
Example 6: Find the absolute maximum value and the
absolute minimum value of the function f (x)= 3 x^4 โ 4 x^3 on
the interval [-1, 2].
Example 7: Find thex coordinate of the absolute extrema of
the function f (x) 3 x^32 x
2 = โ on the interval [-1, 1].
Example 8: An apartment complex has 100 one-bedroom apartment units. Research shows that the monthly profit (in dollars) realized from renting out x apartments is given by
P( x)= โ 12 x^2 + 2256 xโ 48000. How many units should be
rented out to maximize the monthly profit? What is that monthly profit?
Example 9: A company that produces digital cameras wants to minimize its production costs. They estimate that their total monthly cost for producing the camera is given by
C( x)= 0. 0025 x^2 + 80 x+ 10000. Find the average cost
function. Find the level of production that results in the smallest average production cost. Use the second derivative test to verify that you have found a minimum cost.
From this section, you should be able to Identify absolute extrema on ( โโ ,โ)from a graph off. Find absolute extrema on [a,b] algebraically Solve word problems having to do with absolute extrema
