Module 11  Extreme Values and Optimization 
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest 
Lesson 11.1: Finding Extreme Values of a Function 
In this lesson you will explore finding absolute and local extreme values of a function using the TI89. The standard first and second derivative tests will be used to find and identify local extreme values of a function.
Absolute Extreme Values of a Function Let f be a function with domain D. Then f(c) is the (a) absolute maximum value of f on D if and only if f(x) f(c) for all x in D. (b) absolute minimum value of f on D if and only if f(x) f(c) for all x in D. Finding the Absolute Extreme Values of f(x) = x^{2} The graph of f in the figure below suggests that f has no absolute maximum value and has an absolute minimum of 0 at x = 0. Finding the Absolute Extreme Values of f(x) = x^{2} on [2, 3] The graph of f in the window [2, 3] suggests that f has an absolute maximum of 9 at x = 3 and an absolute minimum of 0 at x = 0. This example suggests that existence of absolute extrema depends on the domain of the function. Extreme Value Theorem This important theorem can guide our investigations when we search for absolute extreme values of a function. Theorem 1 If f is continuous on a closed interval [a, b], then f has both an absolute maximum value and an absolute minimum value on the interval. Local Extreme Values of a Function Let c be an interior point of the domain of the function f. Then f has a (a) local maximum at c if and only if f(x) f(c) for all x in some open interval containing c. (b) local minimum at c if and only if f(x) f(c) for all x in some open interval containing c. We can extend the definition of local extrema to endpoints of intervals. A function f has a local maximum or local minimum at an endpoint c if the appropriate inequality holds for all x in some halfopen domain interval containing c. The following theorem helps us find candidates for local extreme values of a function. Theorem 2 If a function has a local maximum value or a local minimum value at an interior point c of its domain and if f ' exists at c, then f ' (c) = 0. We can see from the graph of f(x) = x^{2} above that f has a local minimum of 0 at x = 0. We observed that f has no absolute maximum value but we can also see that f has no local maximum value either. Finding Extreme Values of a Function Because of Theorem 2, we usually need to look at only a few points to find a function's extreme values. Those points consist of interior domain points where f ' = 0 or where f ' does not exist and the domain endpoints which are not covered by the theorem. Critical Points A critical point is an interior point in the domain of a function at which f ' (x) = 0 or f ' (x) does not exist. Find the Extreme Values of f(x) = x^{2} on [2, 3] Refer to the graph of f above.
11.1.1 Draw the graph and find the extreme values of f(x) = x^{2} on [4, 2]. Find the Extreme Values of f(x) = x^{2/3} on [2, 4]
We can see that f ' does not exist at x = 0 and is not equal to 0 anywhere on [2, 4]. The only critical point of f is at x = 0.
Find the value of f at the endpoints. f has a local and absolute minimum of 0 at 0. The value of f at x = 2 is approximately 1.587 and the value at x = 4 is appoximately 2.520. The absolute maximum value of f is approximately 2.520 at x = 4. Find the Extreme Values of f(x) = x^{3} – 2x – 2 cosx
We can see from the graph that f '(x) has a negative zero and a positive zero. Because the sign of f'(x) goes from positive to negative at the negative zero of f '(x), the first derivative test for extreme values tell us that the function f has a local maximum value at this value of x. Similarly, the sign of f '(x) goes from negative to positive at the positive zero of f '(x), so the first derivative test tells us that the function f has a local minimum value at this value of x. We next approximate the zeros of f ' on the home screen as shown in the screen below. The positive zero is approximately 0.559 and the negative zero is approximately 1.263.
According to the second derivative test for extreme values, the value of f at a zero of f ' is
The figure below calculates the value of f " at x = a and x = b, and supports our earlier observations.
You can use these techniques in Lessons 11.211.4. 
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