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Section 4.1 Local Maxima and Minima Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved y How could the local maxima and minima have been found? Figure 4.2: Graph of f (x) = x3 – 9 x2 – 48 x +52 Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Figure 4.6: Changes in direction at a critical point, p: Local maxima or minima Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Consider the graph of f(x). What are the critical points of ? 2 , 5 , 6 , 7 , 8 , 9 What are the local minima of ? 7 , 9 What are the local maxima of ? 2 , 5 , 8 How do the above answers change if the upper hole at 3 is filled in? 3 becomes a critical point How do the above answers change if the lower hole at 3 is filled in? 3 becomes a critical point and a local minimum Find all critical points, local minima, and local maxima of the following functions. = 4 3 + 3 2 − 36 − 5 3 2 = −2 is a local maximum and = is a local minimum g = − 2 ln( 2 + 3) = 1 is a local maximum and = 3 is a local minimum The table records the rate of change of air temperature, H, as a function of time, t, during one morning. When was the temperature a local minimum? Local maximum? t (hours since midnight) dH/dt (◦F/hour) 6 1 A local maximum occurs at 8:00AM A local minimum occurs at 10:00AM. 7 2 8 0 9 −2 10 0 11 3 12 2 Which of the following pieces of information from a daily weather report allow you to conclude with certainty that there was a local maximum of temperature at some time after 10:00 am and before 2:00 pm? (a) Temperature 50◦ at 10:00 am and 50◦ and falling at 2:00 pm. Yes (b) Temperature 50◦ at 10:00 am and 40◦ at 2:00 pm. No, may have been strictly decreasing. (c) Temperature rising at 10:00 am and falling at 2:00 pm. Yes (d) Temperature 50◦ at 10:00 am and 2:00 pm, 60◦ at noon. Yes (e) Temperature 50◦ at 10:00 am and 60◦ at 2:00 pm. No, may have been strictly increasing. If the graph in Figure 4.2 is that of f′(x), which of the following statements is true concerning the function f? (a) The derivative is zero at two values of x, both being local maxima. (b) The derivative is zero at two values of x, one is a local maximum while the other is a local minimum. (c) The derivative is zero at two values of x, one is a local maximum on the interval while the other is neither a local maximum nor a minimum. (d) The derivative is zero at two values of x, one is a local minimum on the interval while the other is neither a local maximum nor a minimum. (e) The derivative is zero only at one value of x where it is a local minimum. Figure 4.2 Problem 7 Graph two continuous functions f and g, each of which has exactly five critical points, the points A-E in Figure 4.12, and which satisfy the following conditions: (a) f (x) → ∞ as x → – ∞ and f (x) → ∞ as x → ∞ (b) g (x) → – ∞ as x → – ∞ and g (x) → 0 as x → ∞ Figure 4.12 Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Problem 36 Assume f has a derivative everywhere and just one critical point, at x = 3. In parts (a) – (d), you are given additional conditions. In each case, decide whether x = 3 is a local maximum, a local minimum, or neither. Sketch possible graphs for all four cases. (a) f ‘(1) = 3 and f ‘(5) = – 1 (b) f (x) → ∞ as x → ∞ and as x → – ∞ (c) f (1) = 1, f (2) = 2, f (4) = 4, f (5) = 5 (d) f ‘(2) = – 1, f (3) = 1, f (x) → 3 as x → ∞ Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved