MATH 1325 – BUSINESS CALCULUS Section 10.2 Applications of the Second Derivative 00 What does f say about f ? Definition: Let f be differentiable on (a;b). 0 The graph of a function, f , is concave upward on the interval (a;b) if f (x) is increasing on (a;b). (If the graph of f lies above all of its tangents on an interval (a;b)) 0 The graph of a function, f , is concave downward on the interval (a;b) if f (x) is decreasing on (a;b). (If the graph of f lies below all of its tangents on an interval (a;b)) Concavity Test (a). If f 00(x) > 0 for all x in (a;b), then f is concave upward on (a;b). (b). If f 00(x) < 0 for all x in (a;b), then f is concave downward on (a;b). A change in concavity has a special name. Definition: A point P on a curve y = f (x) is called if f is continuous there and the curve changes from CU to CD or from CD to CU there. How do we find inflection points? 1. 2. 3. Math 1325 Section 10.2 Continued Ex: Find the intervals where the graph of f (x) = 4x3 + 3x2 − 6x + 1 is concave up, concave down and the inflection points. Then graph. Recall (from Section 10.1): 2 Math 1325 Section 10.2 Continued Ex: Find the intervals where the graph of g(x) = x2 + ln(x2) is concave up, concave down and the inflection points. HINT: Don’t forget about domain! The Second Derivative Test: 1. Compute f 0(x) and f 00(x). 2. Find all critical numbers of f at which f 0(x) = 0. 3. Compute f 00(c) for each critical number found in Step 2. (a) If f 00(c) < 0, then f has a relative maximum at c. (b) If f 00(c) > 0, then f has a relative minimum at c. (c) If f 00(c) = 0 or f 00(c) does not exist, then the test fails. (Inconclusive.) 3 Math 1325 Section 10.2 Continued Ex: Confirm the local extrema for f (x) = 4x3 + 3x2 − 6x + 1 by using the Second Derivative Test. Ex: Find the local extrema for g(x) = ln(x2 + 1) by using the Second Derivative Test. SUMMARY: Signs of f and f 0 Properties of f General Shape of Graph of f f 0(x) > 0 f is increasing f 00(x) > 0 Graph of f is CU f 0(x) > 0 f is increasing f 00(x) < 0 Graph of f is CD f 0(x) < 0 f is decreasing f 00(x) > 0 Graph of f is CU f 0(x) < 0 f is decreasing f 00(x) < 0 Graph of f is CD 4 Math 1325 Section 10.2 Continued Ex: Congress created the alternative minimum tax (AMT) in the late 1970s to ensure that wealthy people paid their fair share of taxes. But because of quirks in the law, even middle-income taxpayers have started to get hit with the tax. The AMT (in billions of dollars) projected to be collected by the IRS from 2001 through 2010 is f (t) = 0:0117t3 + 0:0037t2 + 0:7563t + 4:1 (0 ≤ t ≤ 9) where t is measured in years, with t = 0 corresponding to 2001. (a) Show that f is increasing on the interval (0;9). (b) What does this result tell you about the projected amount of AMT paid over the years in question? (c) Show that f is increasing on the interval (0;9). (d) What conclusion can you draw from this result concerning the rate of growth of the amount of AMT collected over the years in question? 5 Math 1325 Section 10.2 Continued If a company spends money on advertising, the hope is that sales will increase. Sales will increase, rapidly at first, but then the rate of increase will begin to increase less rapidly. This is when the sales are at a maximum value and the function has a point of inflection. This point is called the point of diminishing returns. Ex: An appliance store is selling dryers and decides to invest x thousand in advertising. It estimates that sales can be modeled by the function N(x) = 3x3 − 0:25x2 + 200 (0 ≤ x ≤ 9) NOTE: N0(x) = 9x2 − x3 and N00(x) = 18x − 3x2 N0(x) has a maximum at x = N(x) has an inflection point at x = . The point of diminishing returns is x = . Thus, the rate of change of sales begins to decrease when $ is spent on advertising. 6.