Let f be Continuous on [a; b] and Differentiable on (a; b)

Theorem: Let f be continuous on [a; b] and di erentiable on (a; b). (i) If f 0 (x) > 0 for every x in (a; b), then f is increasing on [a; b]. (ii) If f 0 (x) < 0 for every x in (a; b), then f is decreasing on [a; b]. Exercise 1. If f (x) = x3 + x2 5x 5, (Swok Sec 4.3 Ex 1) (a) nd the intervals on which f is increasing and the intervals on which f is decreasing (b) sketch the graph of f Class Exercise 1. Find the intervals on which the function is increasing or decreasing. (Waits Sec 4.3 #13-19) (a) y = 4x3 + 21x2 + 36x 20 (b) y = 1 =5 y x (c) = 2 + 3 (d) y = 5 x1=3 5e 0 8 e (e) y = e +3 (f) y = 2e 8+5 e 15 e 2x x < 1 (g) f (x) = 2 x2 x  1 x x 4 + 4x3 4x + 1 x x : x x : x First Derivative Test: Suppose that c is a critical number of a continuous function f . (a) If f 0 changes from positive to negative at c, then f has a local maximum at c. (b) If f 0 changes from negative to positive at c, then f has a local minimum at c. (c) If f 0 does not change sign at c (for example if f 0 is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c. Exercise 2. If f (x) = x3 + x2 Exercise 3. If f (x) = x1=3 (8 5x 5, nd the local extrema of f . (Swok Sec 4.3 Ex 2) ), nd the local extrema of f . (Swok Sec 4.3 Ex 3) x Exercise 4. If f (x) = x2=3 (x2 8), nd the local extrema of f . (Swok Sec 4.3 Ex 4) Class Exercise 2. Find the local extreme values for the functions in the previous class exercise. (Waits Sec 4.3 #13-19) (a) y = 4x3 + 21x2 + 36x 20 (b) y = (c) y = 2x1=5 + 3 (d) y = 5 x1=3 5e 0 8 e (e) y = e +3 (f) y = 2e 8+5 e 15 e 2x x < 1 (g) f (x) = 2 x2 x  1 x x 4 + 4x3 4x + 1 x x : x x : x De nition: Let f be di erentiable on an open interval I . The graph of f is (i) concave upward on I if f 0 is increasing on I (ii) concave downward on I if f 0 is decreasing on I Test for concavity: If the second derivative f 00 of graph of f is (i) concave upward on I if f 00 (x) > 0 on I (ii) concave downward on I if f 00 (x) < 0 on I f exists on an open interval I , then the Exercise 5. If f (x) = x3 + x2 5x 5, determine intervals on which the graph of f is concave upward or is concave downward. (Swok Sec 4.4 Ex 1) Exercise 6. If f (x) = sin x, determine where the graph of f is concave upward and where it is concave downward. (Swok Sec Ex 2) Class Exercise 3. Find the intervals on which the function is concave upward or concave downward. (Waits Sec 4.3 #13-19) (a) y = 4x3 + 21x2 + 36x 20 (b) y = x4 + 4x3 4x + 1 (c) y = 2x1=5 + 3 (d) y = 5 x1=3 5e 0 8 e (e) y = e +3 (f) y = 2e 8+5 e 15 e 2x x < 1 (g) f (x) = 2 x2 x  1 x x x : x x : x 1 De nition A point (c; f (c)) on the graph of f is a point of in ection if the following two conditions are satis ed: (i) f is continuous at c. (ii) There is an open interval (a; b) containing c such that the graph is concave downward on (a; c) and concave upward on (c; b), or vice versa. Exercise 7. If f (x) = 12 + 2x2 4 , nd points of in ection of f . (Swok Sec 4.3 Ex 3) x Exercise 8. If f (x) = x5 5x3 , nd the points of in ection of f . (Swok Sec 4.3 Ex 4) Class Exercise 4. Find the points of in ection for the following functions. (Waits Sec 4.3 #13-19) (a) y = 4x3 + 21x2 + 36x 20 (b) y = x4 + 4x3 4x + 1 (c) y = 2x1=5 + 3 (d) y = 5 x1=3 e 5e 0 8 (f) y = 2e 8+5 (e) y = e +3 e 15 e 2x x < 1 (g) f (x) = 2 x2 x  1 x x x : x x : x Second Derivative Test: Suppose that f 00 is continuous near c. (a) If f 0 (c) = 0 and f 00 (c) > 0, then f has a local minimum at c. (b) If f 0 (c) = 0 and f 00 (c) < 0, then f has a local maximum at c. Exercise 9. If f (x) = x5 - 5x3 , nd the local extrema of f . (Swok Sec 4.4 Ex 4) Exercise 10. If f (x) = 1 - x1=3 , nd the local extrema of f , discuss concavity, and nd the points of in ection. (Swok Sec 4.4 Ex 5) Class Exercise 5. Find the intervals on which the function is (i) increasing, (ii) decreasing, (iii) concave up, (iv) concave down, (v) local extreme values, (vi) in ection points . (Waits Sec 4.3 #21-27) p 2 (a) y = xe1=x (b) y = x2 9 x2 (c) y = tan 1 x (d) y = x3=4 (5 x) (e) y = x1=3 (x 4) (f) y = x1=4 (x + 3) 3 2 x 1 (g) y = x 2xx + 2 2