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Section 4.3 Global Maxima and Minima Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Figure 4.30: Global maximum and minimum on an interval domain, a ≤ x ≤ b Figure 4.31: Global maximum and minimum on the entire real line Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Consider the graph of f(x) in the figure. Where is the global maximum of f(x)? 5 Where is a local maximum that is not a global maximum of f(x)? 2 , 8 , 10 Where is the global minimum of f(x)? 3 and 9 Find the global maximum and minimum for = 3 + 5 − 2 on the interval [1, 5]. The global maximum is at = 2.5 The global minimum is at = 5. Find the global maximum and minimum for = − ln on the interval (0, ∞). There is no global maximum. The global minimum is at = 1. On the same side of a straight river are two towns, and the townspeople want to build a pumping station, S. See Figure 4.9. The pumping station is to be at the river’s edge with pipes extending straight to the two towns. Where should the pumping station be located so as to minimize the total length of the pipe? We need to minimize d( x) x 2 1 (4 x)2 16 The global minimum is at = 4/5. The table records the rate of change of air temperature, H, as a function of time, t, one morning. When is it coldest? 11:00AM What temperature was it at that time? Cannot tell from this information If it was 40 ◦F at 7:00AM, estimate the temperature at 11:00AM. 40 − 3 − 0.5 − 1 − 3 = 32.5 ◦F When might there have been an inflection point in H? sometime between 10:00AM and 11:00AM t (hours since midnight) dH/dt (◦F/hour) 7 −5 8 −1 9 0 10 −2 11 −4 Following a dose a of a drug, the average blood pressure of a patient is b. To compute the dose that maximizes the blood pressure, we (a) Find da/db and substitute b = 0. (b) Find db/da and substitute b = 0. (c) Solve for the values of a for which da/db = 0. (d) Solve for the values of a for which db/da = 0. Problem 16 Figure 4.38 shows the rate at which photosynthesis is taking place in a leaf. (a) At what time, approximately, is photosynthesis proceeding fastest for t ≥ 0? 50 days (b) If the leaf grows at a rate proportional to the rate of photosynthesis, for what part of the interval 0 ≤ t ≤ 200 is the leaf growing? When is it growing fastest? t > 0 days; 50 days Figure 4.38 Applied Calculus ,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved