4 3A : Increasing and Decreasing Functions and the First Derivative. Increasing and Decreasing. then

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1 4 3A : Increasing and Decreasing Functions and the First Derivative Increasing and Decreasing! If the following conditions both occur! 1. f (x) is a continuous function on the closed interval [ a,b] and (n other words the graph has no breaks or gaps)! 2. f (x) is differentiable (the derivative is defined) for all x on the open interval ( a,b) (in other words the graph has no sharp points) then! 1. If f (x) > 0 for all x in ( a,b) then the function is Increasing in ( a,b)! 2. If f (x) < 0 for all x in ( a,b) then the function is decreasing in ( a,b)! 3. If f (x) = 0 for all x in ( a,b) then the function is constant in ( a,b)! this means the graph of the function is a horizontal line Strictly Monotonic A function is strictly monotonic in the interval if it is increasing or decreasing for the entire interval. The steps to find the intervals where a function is increasing or decreasing. 1. Find the derivative of the function. 2. Find the critical numbers A. Find the values where the derivative is equal to 0 f (x) = 0 B. Find the values where the derivative is undefined f (x) is undefined 3. Plot the critical numbers on an axis labeled f (x) 4. Determine if each critical number is odd or even. 5. Determine the sign of f (x) for an x value to the right of all critical numbers. If the sign is positive the graph of the function in that interval is increasing. If the sign is negative the graph of the function in that interval is decreasing. Label each interval as increasing or deceasing. 6. Continue to label the intervals to the left as increasing or decreasing. Math ! Page 1! 2015Eitel

2 If the interval to the right of an odd critical number is increasing the interval to the left is decreasing. If the interval to the right of an odd critical number is decreasing the interval to the left is increasing. If the interval to the right of an even critical number is increasing the interval to the left is also increasing If the interval to the right of an even critical number is decreasing the interval to the left is also decreasing 7. List the open intervals where the function is increasing or decreasing using interval notation. Example 1! Example 2 f (x) = 2x 3 3x 2 12x f (x) = 1 3 x3 + 3x 2 8x f (x) = 6x 2 6x 12 ( ) f (x) = 6 x 2 x 2 f (x) = 6 x 2 ( ) ( x +1) critical numbers where f (x) = 0 x 2 = 0 x +1= 0 x = 2 x = 1 f (x) = x 2 + 6x 8 f (x) = x 2 ( ) ( x 4) critical numbers where f (x) = 0 x 2 = 0 x 4 = 0 x = 2 x = 4 critical numbers where f (x) undefined. NONE! critical numbers where f (x) is undefined NONE Math ! Page 2! 2015Eitel

3 f (x) is increasing on(, 1) ( 2, ) f (x) is decreasing on( 1,2 )! f (x) is increasing on( 2,4 ) f (x) is decreasing on(,2) ( 4, )! Example 3 f (x) = f (x) = x2 x 1 x(x 2) ( x 1) 2 critical numbers where f (x) = 0 x = 0 x 2 = 0 x = 0 x = 2 critical numbers where f (x) is undefined x 1= 0 x = 1 Math ! Page 3! 2015Eitel

4 00 f (x) is increasing on (,0) ( 2, ) f (x) is decreasing on( 0,1 ) ( 1,2 ) The First Derivative Test for Maximums and Minimums then there is at least one real number c in the open interval ( a,b) where f (c) = 0 or in other words the graph has a horizontal tangent line at the point ( a,b) Extrema General Definition of Extrema Let a function f be defined on an Interval containing c, then: ( c, f (c) ) in the he open interval 1. f (c) is the maximum value of the function in an interval if 2. f (c) is the minimum value of the function in the interval If f (c) f (x) for ALL x in the interval f (c) f (x) for ALL x in the interval Math ! Page 4! 2015Eitel

5 The maximum and minimum y values on the interval are called the extreme values or Extrema (plural) of the function on the interval. Definition of Relative Extrema (Local Maximum and Local Minimums) 1. Let a function f be defined on an OPEN INTERVAL (a,b) and let f (c) be the maximum value for the function on the OPEN INTERVAL (a,b). Then f (c) ic called a relative maximum value of the function for that interval (a,b). We use the term local maximum because it is only the maximum value for the open interval (a,b) Other open intervals like (a 1,b 1 ) may also have a local maximum at an x value in that interval. In fact, a function may have several local maximums on several different open intervals 2. Let a function f be defined on an OPEN INTERVAL (a,b) and let f (c) be the minimum value for the function on the OPEN INTERVAL (a,b). Then f (c) ic called a relative minimum value of the function f for that interval (a,b). We use the term local minimum because it is only the minimum value for the open interval (a,b). Other open intervals like (a 1,b 1 ) may also have a local minimum at an x value in that interval. In fact, a function may have several local minimum on several different open intervals (Endpoint Extrema) Endpoints on Closed Intervals Math ! Page 5! 2015Eitel

6 Closed Intervals have endpoints. Each endpoint is either a minimum value or a maximum value for that neighborhood but are not considered relivite mi,imu, or relative maximum. These local extrema are called Endpoint Extrema. If the function f (x) is a continuous function on the closed interval [ a,b] ( a, f (a) ) and ( b, f (b) ) are called Endpoint Extrema. then the endpoints The values of f (a) and f (b) are either a local minimum or a local maximum. Open Intervals have end circles that CANNOT be local extrema. The Absolute Maximum Value of a Function and The Absolute Minimum Value of a Function 1. f (c) is the Absolute Maximum value of the function f (x) if f (c) f (x) all values of x 2. f (c) is the Absolute Minimum value of the function f (x) if f (c) f (x) all values of x Note: lim If the x a F(x) = for any value of a in the domain of f (x) then there is NO Absolute Maximum Value for the function lim If the x a F(x) = for any value of a in the domain of f (x) then there is NO Absolute Minimum Value for the function Math ! Page 6! 2015Eitel

7 Theorem 4.1 The Extreme Value Theorem (Existence) If f (x) is a continuous function on the closed interval [ a,b] then f (x) has at least one absolute maximum value and at least one absolute minimum value on the closed interval [ a,b]. These values occur at an x value on the open interval ( a, b) or at an endpoint of the closed interval. While the Extreme Value Theorem tells us that an absolute maximum and an absolute maximum exist, it does not tell us where to find these points. That problem is solved by the introduction of Critical Numbers. Definition of Critical Number Given a function f (x) and a real number c If f (c) = 0 or if f (c) is undefined at x = c then c is a critical number for the function. Critical Numbers are the value(s) of x where the derivative of the function equals zero OR where the derivative of the function is undefined. Theorem 4 2 The Relative Extrema for the function can occur only at critical x values for the function or at the endpoints of closed intervals. The y value for each critical x number or endpoint of a closed interval is a possible extrema for the function. Math ! Page 7! 2015Eitel

8 If x = c is a critical number the y value for the point ( c, f (c) ) is f (c) The value of f (c) is a possible extrema for the function. The endpoints of a closed interval [ a,b] are ( a, f (a) ) and and the y values for the endpoints are f (a) and f (b) ( b, f (b) ) f (a) and f (b) also possible extrema for the function. There are no other possible x values where possible extrema for the function can occur. The Steps for Finding Relative Extrema on a Closed Interval for a continuous function If a function f is continuous on a closed interval [ a,b] then the following steps will find the Local Extrema Step 1. Find all critical x values. Find al the values of x where f (x) = 0 or where f (x) is undefined. Step 2. If the critical x values are x 1, x 2..., x n then the y values for each critical x value are ( ), f ( x 2 )..., f ( x n ) These y values are the possible Relative Extrema f x 1 Step 3. The function is defined on a closed interval [ a,b]. Find the y values for f (a) and f (b) Step 3. The largest y value from f ( x 1 ), f ( x 2 )..., f ( x n ), f (a) and f (b) is a maximum value for the function on the closed interval.! The smallest y value from f ( x 1 ), f ( x 2 )..., f ( x n ), f (a) and f (b) is a minimum value for the function on the closed interval. Math ! Page 8! 2015Eitel

9 Optimization problems are one of the most important applications of differential calculus because we often want to know when the output of a function is at its maximum or minimum. In such problems there may be a largest or smallest output value over the entire input interval of interest or within a local neighborhood of an input value. Both absolute and local maximum and minimum values are of interest in many contexts. Absolute Extreme Values of a Function When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum, as defined below. Let f be a function with domain D and let c be a fixed constant in D. Then the output value f(c) is the absolute maximum value of f on D if and only if f(x) f(c) for all x in D. absolute minimum value of f on D if and only if f(c) f(x) for all x in D. Absolute Extreme Values - An Example The domain of f(x) = x2 is all real numbers and the range is all nonnegative real numbers. The graph in the figure below suggests that the function has no absolute maximum value and has an absolute minimum of 0, which occurs at x = 0. Math ! Page 9! 2015Eitel

10 [-5, 5, 1] x [-2, 10, 1] The Absolute Extreme Values on a Restricted Domain If the domain of f(x) = x2 is restricted to [-2, 3], the corresponding range is [0, 9]. As shown below, the graph on the interval [-2, 3] suggests that f has an absolute maximum of 9 at x = 3 and an absolute minimum of 0 at x = 0. Math ! Page 10! 2015Eitel

11 The two examples above show that the existence of absolute maxima and minima depends on the domain of the function. Extreme Value Theorem Theorem 1 below is called the Extreme Value theorem. It describes a condition that ensures a function has both an absolute minimum and an absolute maximum. The theorem is important because it 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. This theorem says that a continuous function that is defined on a closed interval must have both an absolute maximum value and an absolute minimum value. It does not address how to find the extreme values. Local Extreme Values of a Function One of the most useful results of calculus is that the absolute extreme values of a function must come from a list of local extreme values, and those values are easily found using the first derivative of the function. Local extreme values, as defined below, are the maximum and minimum points (if there are any) when the domain is restricted to a small neighborhood of input values. Let c be an interior point of the domain of the function f. Then the function f has a local maximum at c if and only if f(x) f(c) for all x in some open interval containing c. local minimum at c if and only if f(c) f(x) for all x in some open interval containing c. Endpoints as Local Extrema Math ! Page 11! 2015Eitel

12 The definition of local extrema given above restricts the input value to an interior point of the domain. The definition can be extended to include endpoints of intervals. A function f has a local maximum or local minimum at an endpoint c of its domain if the appropriate inequality holds for all x in some half-open interval contained in the domain and having c as its one endpoint. It is clear from the definitions that for domains consisting of one or more intervals, any absolute extreme point must also be a local extreme point. So, absolute extrema can be found by investigating all local extrema. Candidates for Local Extreme-Value Points Theorem 2 below, which is also called Fermat's Theorem, identifies candidates for local extremevalue points. 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. Finding Extreme Values of a Function Theorem 2 says that if a function has a first derivative at an interior point where there is a local extremum, then the derivative must equal zero at that point. It does not say that every point where the first derivative equals zero must be a local extremum. Because of Theorem 2, only a few points need to be considered when finding a function's extreme values. Those points consist of interior domain points where f ' (x)= 0, interior domain points where f ' does not exist, and the domain's 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 ' does not exist. So the only possible candidates for the x-coordinate of an extreme point are the critical points and the endpoints. Math ! Page 12! 2015Eitel

13 Finding the Extreme Values Using Calculus Techniques Find the local and absolute extreme values of f(x) = x2 on the closed interval [-2, 3] using calculus. Theorem 1 applies here, so we know for certain that this function must have absolute extrema on this domain. Note the following: f ' (x) = 2x, which is zero only at x = 0 and exists at all values of f in [-2, 3]. Therefore, x = 0 is the only critical point of f. The values of f at the endpoints are f(-2) = 4 and f(3) = 9. By comparing the output values when x = -2, x = 0, and x = 3, the absolute extrema may be determined. f has a local minimum of 0 at x = 0, which is also an absolute minimum. f has a local maximum of 4 at x = -2 and a local maximum of 9 at x = 3. The absolute maximum of f is 9. Review the graph of the function on the restricted domain. The graph supports the above results. [-2, 3, 1] x [-2, 10, 1] Math ! Page 13! 2015Eitel

14 Find the extreme values of f(x) = x2 on [-4, 2] using calculus techniques and then support your answers by sketching the graph. Click here for the answer. Calculus techniques produce results that may be supported by graphs, and graphs can guide in the discovery of extreme values, as shown in the next example. Extreme Values of f(x) = x2/3 on [-2, 4] Find the extreme values of f(x) = x2/3 on the restricted domain [-2, 4] by viewing the graph and then using calculus techniques. Enter X^(2/3) in Y1. Display the graph in a [-2, 4,1] x [-1, 3,1] window. The function appears to have an absolute minimum near x = 0 and two local maximums, which occur at the endpoints of the restricted domain. The absolute maximum occurs at the right endpoint of the restricted domain. Now determine the extreme points using calculus techniques. Use the power rule to find f ': The derivative,, is not equal to 0 anywhere on [-2, 4], so no critical point comes from that condition, but f ' does not exist at x = 0, which implies that x = 0 is a critical point. Therefore, the only critical point of f occurs at x = 0. Math ! Page 14! 2015Eitel

15 Use the Graph screen's Value feature to compute the values of f at the critical point and at the endpoints of the restricted domain [-2, 4]. From the graph of f press [CALC] and select 1:value. Evaluate f at x = -2, x = 0, and x = 4 by entering -2, 0, and 4, respectively. Math ! Page 15! 2015Eitel

16 The extreme values can be summarized as follows: f has a local and absolute minimum of 0 at 0. The value of f at x = -2 is approximately and the value at x = 4 is approximately Each is a local maximum value. The absolute maximum value of f is approximately at x = 4. Extreme Values of In the previous examples, we have been dealing with continuous functions defined on closed intervals. In such a case, Theorem 1 guarantees that there will be both an absolute maximum and an absolute minimum. In this example, the domain is not a closed interval, and Theorem 1 doesn't apply. The extreme values of may be found by using a procedure similar to that above, but care must be taken to ensure that extrema truly exist. Notice that the domain of f is (-2, 2) because the radicand must be non-negative and the denominator must be non-zero. Graph y = 1 4 x 2 in a [-4, 4, 1] x [-2, 4, 1] viewing window. Math ! Page 16! 2015Eitel

17 The graph suggests there is an absolute minimum of about 0.5 at x = 0. There also appear to be local maxima of about 2.5 when x = -2 and x = 2. However, f is not defined at x = -2 and x = 2, so they cannot be local maxima. Calculus techniques require that the endpoints of the domain and critical points must be identified. y = The domain of f is (-2, 2), an open interval, so there are no endpoints. Critical points are determined by using the derivative, which is found with the Chain Rule. x ( 4 x 2 ) 3/2 The derivative is 0 at x = 0 and it is undefined at x = -2 and x = 2. Because -2 and 2 are not in the domain of f, the only critical point is x = 0. As x moves away from 0 in either direction, the denominator of f(x) gets smaller and f(x) gets larger. Thus, f has an absolute minimum of 0.5 at x = 0. There are no absolute maximum points. This does not violate the Extreme Value theorem because the function is not defined on a closed interval. Since an absolute maximum must occur at a critical point or an endpoint, and x = 0 is the only such point, there cannot be an absolute maximum.!! Critical Values That Are Not Extrema A function's extreme points must occur at critical points or endpoints, however not every critical point or endpoint is an extreme point. The following graphs of y = x3 and illustrate critical points at x = 0 that are not extreme points Math ! Page 17! 2015Eitel

18 ! y = x 3 in a [-3, 3 1] x [-2, 2, 1] window! i y = 3 x = x 1/3 n a [-3, 3, 1] x [-2, 2, 1] window Notice that the derivative of y = x 3 is y = 3x2 and the derivative of y = 3 x = x 1/3 y = is. 1 3x 2/3 The first derivative of y = x 3 is zero when x = 0 and the first derivative of y = 3 x does not exist at x = 0. Although x = 0 is a critical point of both functions, neither has an extreme value there. In addition to finding critical points using calculus techniques, viewing the graph of a function should help identify extreme values. In these two examples, note that the first derivative is positive on both sides of x = 0. In lesson 13.2 we will use the First Derivative Test, where the sign of the derivative on either side of a critical point is used to determine whether the critical point is a local maximum, a local minimum, or neither. Math ! Page 18! 2015Eitel

19 The First Derivative Test The First Derivative Test consists of three parts: using the first derivative when testing for local maxima, when testing for local minima, and when identifying critical points that are neither. Additionally, the nature of endpoints may also be determined by using the first derivative. Collectively, the parts are called the first derivative test. Each part is discussed below and is then stated formally. The First Derivative Test to Find a Local Maximum Recall that the first derivative represents the slope of the tangent line to the curve. The diagram below shows a function near a critical point, which is represented by the vertical line. The plus and minus signs indicate where the function's first derivative is positive and where it is negative. Notice that when the first derivative is positive, the slopes are positive and the function is increasing. When the first derivative is negative, the slopes are negative and the function is decreasing. If the derivative changes from positive to negative at the critical point, the critical point is a local maximum Part I If the first derivative is positive to the left of a critical point and negative to the right of the critical point, then f has a local maximum at the critical point. The First Derivative Test to Find a Local Minimum The first derivative of the function shown below has negative values left of the critical point and positive values right of the critical point, as indicated by the diagram. The critical point is a local minimum. Math ! Page 19! 2015Eitel

20 Part II If the first derivative is negative to the left of a critical point and positive to the right of the critical point, then f has a local minimum at the critical point The First Derivative Test for Neither As discussed at the end of Lesson 13.1, x = 0 is a critical point of f(x) = x3 but there is no local extremum there. It was noted that the first derivative (slope of the tangents) is positive on both sides of the critical point, so the sign of the derivative does not change from one side of x = 0 to the other. The general statement about the sign of the derivative at non-extreme points is given below. Part III If the first derivative has the same sign on both sides of the critical point, then the function does not have a local maximum or a local minimum at the critical point. Extrema at Endpoints A function's first derivative can only be evaluated on one side of an endpoint of a function's domain. Whether an endpoint is a local maximum or minimum is determined by the sign of the first derivative when evaluated just inside the domain near the endpoint. Local Extrema at a Left Endpoint Math ! Page 20! 2015Eitel

21 If the first derivative is positive to the right of a left endpoint, the function has a local minimum at the left endpoint. If the derivative is negative to the right of a left endpoint, the function has a local maximum at the left endpoint. The diagrams below illustrate a left endpoint that is a local minimum and a left endpoint that is a local maximum. Recall that a function is increasing when the slope of a tangent line is positive and decreasing when the slope of a tangent line is negative. The plus and minus signs indicate the sign of the first derivative, which also represents the slope of the tangent lines and indicate whether the function is increasing or decreasing. The First Derivative Test at a Right Endpoint If the first derivative is negative to the left of a right endpoint, the function has a local minimum at the right endpoint. If the derivative is positive to the left of a right endpoint, the function has a local maximum at the right endpoint. The diagrams below illustrate a right endpoint that is a local minimum and a right endpoint that is a local maximum. Again, the plus and minus signs indicate the sign of the first derivative and indicate whether the function is increasing or decreasing. Math ! Page 21! 2015Eitel

22 Math ! Page 22! 2015Eitel