The Extreme Value Theorem (IVT)

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Erik Francis
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1 old school 1 Extrema If f(c) f(x) (y values) for all x on an interval, then is the (value) of f(x) (the function) on that interval. If f(c) f(x) (y-values) for all x on an interval, then is the (value) of f(x) (the function) on that interval. The Extreme Value Theorem (IVT) If f(x) is on a interval [a, b], then f(x) has both a and a minimum on [a, b]. Where, if anywhere, do the graphs indicate an absolute extrema? Does the EVT apply?

2 old school 2 Relative extrema A relative or extremum: Look back at the graphs we just looked at and find the relative max and mins. What is true, calculusly, at each of these points? Critical numbers: How does one find the absolute min and max y-value of a function on a closed interval? Find the absolute extrema of f(x) = 2x 3x 2/3 on [-1, 3]. 1. Find all, if any, numbers on the interval. 2. Find the y-value for each critical number and for each. 3. y-values, The is the max and the smallest the minimum. Page , 15, 17, 21, 25, 29, 33, 41, 63-66

3 old school 3 Rolle s Theorem If f(x) is continuous on and on (a, b), and if f(a) = ( = 0), then there at least one number c on such that =. Rolle s theorem is another theorem, that does not tell us to find the c, just that one exists. If Rolle s Theorem applies, find the c(s) guaranteed by Rolle s Theorem for f(x) = x 4 2x 2 on [-2, 2]. Besides f '(c) = 0 Rolle s is saying:

4 old school 4 Mean Value Theorem If f(x) is on [a, b] and differentiable on, then there at least one number c on such that f ' (c) =. A picture is worth a words: Find the c guaranteed by the MVT for 4 y on [1, 4]. x Besides saying that f '(c) = Pg , 15, 19, 35, 41, 45

5 old school 5 Increasing/decreasing Definitions: If b > a and if f(b) > f(a) for all values of x on an interval then the function is said to be on that interval. If b > a and if f(b) f(a) for all values of x on an interval then the function is said to be on that interval. Facts: It is the definition of increasing, the above is, but a fact about an increasing interval is that is. For decreasing intervals, is. What is true graphically and calculusly when a graph changes from increasing to decreasing? The f (x) test Find all, if any, intervals of increasing or decreasing and find all, if any relative maximum and minimum points for 3 f (x) x x

6 old school 6 The first derivative test is a test to determine if a point is a maximum, minimum or. Use the fist derivative test and find all, if any, inc. and dec. intervals and all, if any, relative max and mins for f(x) = (x 2-4) 2/3. Page , 29, 35, 39 Not the f (x) test What is happening graphically when f (x) = 0? Is the graph increasing or decreasing? If a derivative is a rate of change, then f (x) = 0 says that f (x) is. What is happening graphically when f (x) is positive? or negative?

7 old school 7 The f tells us if the graph is up or down. At some point a graph will change from being concave up to concave down, or. The point at which this change occurs is known as an point. The definition of an inflection point is the point where a graph A fact about an inflection point is that at an inflection point = 0 or undefined. Find all, if any, intervals where f(x) is concave up and down and find all, if any, points of inflection for f(x) = -x 3 +3x 2-2 NOTE: What we just did is called the second derivative test. It isn t called but we can call it There is by the way a derivative test. This just it. If the sign of the f ' does change around a critical number then that point is an point. Two types of inflection points: Page , 19, 21

8 old school 8 The f (x) test? f (2) = 5 f (2) = 0 f (2) = 5 sup at 2? The tests to see if a point is relative min or max. f(2) = 3 f (2) = 0 f(1) = 5 f(5) = 7 If 2 is the only x for which f ' (x) = 0, sup at 2? The tests to see if a point is relative min or max. If you are not testing to see if a is a min, max or, then you are not doing the f(x), f (x) or tests. Page Page , 35, 37

9 old school 9 The full package curve sketch Find all, if any, intervals where f(x) is increasing/decreasing. Find all, if any, relative maximum or minimums. Find all, if any, intervals where f(x) is concave up/down. Find all, if any, inflection points. Use all this to sketch the graph of f(x). 5 3 f (x) 3x 5x Full package y = sin 2 (x) [0, 2 ] y 2x 2 x 1

4.3 Finding Local Extreme Values: First and Second Derivatives

Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan 4.3 Finding Local Extreme Values: First and Second Derivatives Recall that a function f(x) is said to be increasing (respectively decreasing)