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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVES Find relative extrema of a continuous function using the First-Derivative Test. Shi,Chen

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs DEFINITIONS: A function f is increasing over I if, for every a and b in I, if a < b, then f (a) < f (b). (If the input a is less than the input b, then the output for a is less than the output for b. A function f is decreasing over I if, for every a and b in I, if a f (b). (If the input a is less than the input b, then the output for a is greater than the output for b.)

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A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right. inc dec

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The slope of the tan line is positive when the function is increasing and negative when decreasing

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THEOREM 1 If f (x) > 0 for all x in an interval I, then f is increasing over I. If f (x) < 0 for all x in an interval I, then f is decreasing over I. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

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Find the intervals where f is increasing and decreasing

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Since f (x) = 2x+5 it follows that f is increasing when 2x+5>0 or when x>-2.5 which is the interval

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We use a similar method to find the interval where f is decreasing. 2x+5<0 gives x < -2.5 or the interval

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Find the intervals where the function is increasing and decreasing

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A product has a profit function of for the production and sale of x units. Is the profit increasing or decreasing when 100 units have been sold?

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Suppose a product has a cost function given by Find the average cost function. Over what interval is the average cost decreasing?

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DEFINITION: A critical value of a function f is any number c in the domain of f for which the tangent line at (c, f (c)) is horizontal or for which the derivative does not exist. That is, c is a critical value if f (c) exists and f (c) = 0 or f (c) does not exist. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

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DEFINITIONS: Let I be the domain of f : f (c) is a relative minimum if there exists within I an open interval I 1 containing c such that f (c) f (x) for all x in I 1 ; and F (c) is a relative maximum if there exists within I an open interval I 2 containing c such that f (c) f (x) for all x in I 2.

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 2 If a function f has a relative extreme value f (c) on an open interval; then c is a critical value. So, f (c) = 0 or f (c) does not exist.

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 3: The First-Derivative Test for Relative Extrema For any continuous function f that has exactly one critical value c in an open interval (a, b); F1. f has a relative minimum at c if f (x) < 0 on (a, c) and f (x) > 0 on (c, b). That is, f is decreasing to the left of c and increasing to the right of c.

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 3: The First-Derivative Test for Relative Extrema (continued) F2. f has a relative maximum at c if f (x) > 0 on (a, c) and f (x) < 0 on (c, b). That is, f is increasing to the left of c and decreasing to the right of c. F3. f has neither a relative maximum nor a relative minimum at c if f (x) has the same sign on (a, c) and (c, b).

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1: Graph the function f given by and find the relative extrema. Suppose that we are trying to graph this function but do not know any calculus. What can we do? We can plot a few points to determine in which direction the graph seems to be turning. Lets pick some x-values and see what happens.

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (continued):

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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (continued): We can see some features of the graph from the sketch. Now we will calculate the coordinates of these features precisely. 1 st find a general expression for the derivative. 2 nd determine where f (x) does not exist or where f (x) = 0. (Since f (x) is a polynomial, there is no value where f (x) does not exist. So, the only possibilities for critical values are where f (x) = 0.)

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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (continued): These two critical values partition the number line into 3 intervals: A (–, –1), B (–1, 2), and C (2, ).

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Example 1 (continued): 3 rd analyze the sign of f (x) in each interval. Test Valuex = –2x = 0x = 4 Sign of f (x) +–+ Result f is increasing on (–, –1] f is decreasing on [–1, 2] f is increasing on [2, ) Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

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2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (concluded): Therefore, by the First-Derivative Test, f has a relative maximum at x = –1 given by Thus, (–1, 19) is a relative maximum. And f has a relative minimum at x = 2 given by Thus, (2, –8) is a relative minimum.

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