Critical Points, Local Maxima, and Local Minima Find all critical points, local minima, and local maxima of the following functions. f ( x) = 4 x 3 + 3 x 2 − 36 x − 5 What are the critical points of ? What are the local minima of ? What are the local maxima of ? How do the above answers change if the upper hole at 3 is filled in? How do the above answers change if the lower hole at 3 is filled in? The table records the rate of change of air temperature, H, as a function of hours since midnight, t, during one morning. t dH / dt 6 7 8 9 10 11 12 1 2 0 -2 0 3 2 When was the temperature a local minimum? Local maximum? g ( x) = x − 2 ln( x 2 + 3) Critical Points, Local Maxima, and Local Minima 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. (b) Temperature 50◦ at 10:00 am and 40◦ at 2:00 pm. 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 → ∞ (c) Temperature rising at 10:00 am and falling at 2:00 pm. (d) Temperature 50◦ at 10:00 am and 2:00 pm, 60◦ at noon. (e) Temperature 50◦ at 10:00 am and 60◦ at 2:00 pm. If the graph is that of f '( x) , state true or false for the following statements about 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 max. and one is a local min. (c) The derivative is zero at two values of x, one is a local max. while the other is neither a local max. nor a min (d) The derivative is zero at two values of x, one is a local min. while the other is neither a local max. nor a min. (e) The derivative is zero only at one value of x where it is a local min. 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 →∞

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