4.3 Finding Local Extreme Values: First and Second Derivatives

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1 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) in an interval I if f(x 1 ) < f(x 2 ) (respectively f(x 1 ) > f(x 2 )) whenever x 1 < x 2 where x 1 and x 2 are in the domain of f. The following result provides a way for testing whether a function is increasing or decreasing: Increasing/Decreasing test If f (x) > 0 on an open interval I then f(x) is increasing on I. If f (x) < 0 on an open interval I then f(x) is decreasing on I. Example Consider the function f(x) = x 3 9x 2 48x Find the intervals where the function is increasing/decreasing. Finding the first derivative we obtain f (x) = 3x 2 18x 48 = 3(x 8)(x+2). Constructing the chart of signs as shown in Figure Figure we see that f(x) is increasing on (, 2) (8, ) and decreasing on ( 2, 8) Next we will discuss a procedure for finding local extrema of a function based on the first derivative which we call the first derivative test. First-Derivative Test Suppose that a continuous function f has a critical point at p with f (p) = 0. If f changes sign from negative to positive at p, then f has a local minimum at p. If f changes sign from positive to negative at p, then f has a local maximum at p. See Figure

2 Figure Example (a) Find the local extrema of the function f(x) = x 3 9x 2 48x (b) Find the local extrema of the function f(x) = sin x + e x, x 0. (a) Using the chart of signs of f discussed in Example 4.3.1, we find that f(x) has a local maximum at x = 2 and a local minimum at x = 8. (b) Finding the derivative to obtain f (x) = cos x+e x. But for x 0, 1 e x. Since 1 cos x 1, adding the two inequalities we see that 0 cos x + e x. This implies that f does not change sign for x 0. Therefore, there are no local maxima. The only local minimum occurs at the point (0, 1) As the first derivative provides information about whether a function is increasing or decreasing, the second derivative can be used to provide informtion about whether the graph opens up or down. We say that a graph is concave up in an interval I if all the tangent lines on I lie below the graph. In the case all the tangent lines on I lie above the graph, we say that the graph is concave down. We have the following Concavity test If f (x) > 0 on an open interval I then f(x) is concave up on I. If f (x) < 0 on an open interval I then f(x) is concave down on I. Example Consider the function f(x) = x 3 9x 2 48x Find the intervals where the function is concave up or down. 2

3 Finding the second derivative, we obtain f (x) = 6x 18 = 6(x 3). So, f(x) is concave up on (3, ) and concave down on (, 3) We have seen that a local extremum is a point where the first derivative changes sign. Next, we will discuss points where the second derivative changes sign. That is, the points where the graph of the function changes concavity. We call such points points of inflection. How do you find the points of inflection? Well, since f changes sign on the two sides of an inflection point then it makes sense to say that points of inflection occur at points where either the second derivative is 0 or undefined. Example Find the point(s) of inflection of the function f(x) = xe x. Using the product rule to obtain f (x) = e x xe x. Using the product rule for the second time we find f (x) = e x (x 2). Thus, a candidate for a point of inflection is x = 2. Since f (x) > 0 for x > 2 and f (x) < 0 for x < 2, x = 2 is a point of inflection Remark We have seen that not every value of x where the derivative is zero or undefined is a local maximum or minimum. The same thing applies for points of inflection. That is, it is not always true that if the second derivative is 0 or undefined then automatically you have a point of inflection. It is critical that f changes sign at such a point in order to have a point of inflection. Example Consider the function f(x) = x 4. Show that f (0) = 0 but 0 is not a point of inflection. The second derivative is given by the formula f (x) = 12x 2. Clearly, f (0) = 0. Since f (x) 0, that is, f (x) does not change sign then 0 is not a point of inflection Example Graph a function with the following properties: f has a critical point at x = 4 and an inflection point at x = 8. 3

4 f < 0 for x < 4 and f > 0 for x > 4. f > 0 for x < 8 and f < 0 for x > 8. The graph is given in Figure Figure One can use concavity to decide whether a critical point is a maximum or a minimum. Second-Derivative Test Let f be a continuous function such that f (p) = 0. if f (p) > 0 then f has a local minimum at p. See Figure if f (p) < 0 then f has a local maximum at p. See Figure if f (p) = 0 then the test fails. In this case, it is recommended that you use the first derivative test. Figure Example Use the second derivative test to find the local extrema of the function f(x) = x 3 9x 2 48x

5 The second derivative of f(x) is given by f (x) = 6(x 3). The critical numbers are 2 and 8. Since f ( 2) = 30 < 0, x = 2 is a local maximum. Since f (8) = 30 > 0, x = 8 is a local minimum Example Find the local extrema of the function f(x) = x 4. Let s try and find the local extrema by using the second derivative test. Since f (x) = 4x 3, x = 0 is the only critical number. Since f (x) = 12x 2, f (0) = 0. So the second derivative test is inconclusive. Now, using the first derivative test, we see that f (x) changes sign from negative to positive at x = 0. Thus, x = 0 is a local minimum 5