# maximum and minimum of the functions

1. Find the absolute maximum and minimum of the functions on the given interval and identify where they occur. a) f(x) = – in (1/2,2] b) f(t) = t – 5], t in (4,7] c) f(1) == in (-2,1] 2. a) Sketch a differentiable function that is increasing and concave up on (-00,00). What can be said about f’ and f”? b) Sketch a differentiable function that is increasing and concave down on (-00,00). What can be said about f’ and f”? c) Sketch a differentiable function that is decreasing and concave up on (-00,00). What can be said about f’ and f”? d) Sketch a differentiable function that is decreasing and concave down on (-00,00). What can be said about f’ and “? 3. Sketch a function that satisfies all the following criteria: i. Differentiable on (-00,00) iv. Decreasing and concave down on (0,2) ii. Increasing and concave up on (-00,-2) v. Decreasing and concave up on (2,4) iii. Increasing and concave down on (-2,0) vi. Decreasing and concave down (4,00) a) Suppose c is a critical point of f, what do know about f'(c). b) How does knowing the critical points of a function help understand its graph? 5. Determine whether the following are true of false. Provide an explanation or counterexample a) The function f(x) = Vt has an absolute maximum. b) The function f(x) = Vi has an absolute maximum on the interval (0,1). 6. Determine whether the following are true of false. Provide an explanation or counterexample a) If a function has an absolute maximum, the function must be continuous on a closed interval. b) If a function has the property f'(2) = 0, then f has a local min or local max at 1 = 2. c) Absolute extreme values on an interval always occur at a critical point or an endpoint of the interval 1 d) If a function has the property that f'(3) does not exist, then I = 3 is a critical point of f. 7. Provide an example of each of the following: a) f'(0) = 0 but f’ does not change sign at 1 = 0. b) f(0) = 0 but the second derivative test is inconclusive. c) f(x) has an inflection point at 2 = 0. d) f”0) = 0 but there is not an inflection point at < = 0. 8. Determine whether the following are true of false. Provide an example or counterexample (sketches are ok). a) f’ can remain constant while f” changes. b) f” can remain constant while f’ changes. c) f’ and f” can both remain constant. d) f’ and f” can both change. e) f can only change concavity if f’ (t) 0. f) f can have a critical point without changing concavity. 9. An apartment building has 100 rental units. The management company knows from experience that if they charge \$800/month, all apartments will be occupied. A survey suggests that, on average, one additional apartment will remain vacant for every \$10 increase in rent. What rent should the man- agement company charge to maximize revenue. 10. A boat leaves a dock at 2:00 P.M. and travels due south at a speed of 20 km/hr. Another boat has been heading due east at a speed of 15 km/hr and reaches the dock at 3:00 P.M. At what time were the two boats closest together. 11. For the following functions locate the critical values, intervals where the function is increas- ing/decreasing, local maxima and minima (if any) and absolute maxima and minima (if any). 1. f(x) = 2x – x? on (-1,2] 2. f(x) = 25 – ron (-5,5) 12. For the following functions locate the critical values, inflection points, intervals where the function is increasing/decreasing, intervals where the concavity is up/down, local maxima and minima (if any) and absolute maxima and minima (if any). 1. f(x) = x(x + 2) on R 2. f(x) = /(- x) on R 3. f(x) = on its domain
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