Question 15. Version 1*/1. Score: 111 Suppose at) and f'(x) are continuous but restricted to the interval 0 S m S 20, and assume the values of f'(m) are as shown. For each value, determine whether there is a local maximum, local minimum, or nothing. "II-n "\"3! At .7; = 0, you guarantee :DV 0" At .7; = 5, you guarantee (:V 0\" nothing At a: = 10, you guarantee (:V 0" At .7; = 15, you guarantee 2\" 0" At a: = 20, you guarantee (:V 0" Show Detailed Solution First, we'll check the endpoints. At m : O, the derivative is positive, and since the graph begins at m : 0, this means the graph will be rising from the point, so a: = 0 will be a local minimum. At m : 20, the derivative is negative, and since the graph ends at a: : 20, this means the graph will be falling to the point, so m : 0 will be a local minimum. On the interior of the interval, anywhere the derivative is not equal to 0 will not correspond to a local maximum or minimum value, so the only place where a local minimum or maximum value might occur is at :8 : 5. We see that at a: = 0, f'(0) : 1, so the graph is rising until it reaches x = 5. Meanwhile, f'(10) : 4, so the graph is rising after a: = 5. This means that a: = 5 will be a nothing. Question 13. Version 1*/2. Score: 111 v Suppose at) and f'(a:) are continuous everywhere, and have the following values: :1: O 10 2O 30 4O f'(9:) 20 9 18 i4 3 Based on this, determine what you are willing to guarantee. Between 93 = 0 and 2: = 10 you guarantee (:V cr' Between :1: = 10 and :1: = 20 you guarantee :)V 0\" Between .7; = 20 and :L- = 30 you guarantee 2\" (3' Between a: = 30 and a: = 40 you guarantee :D" 0" Show Detailed Solution This solution is for a similar problem, not your specific version Suppose f(.'B) and f'(m) are continuous everywhere, and have the following values: TWWWWW m) FWWWW Since f'(:1:) is continuous, and f'(10) : 9, f'(20) : 3 , the intermediate value theorem guarantees that f'(cl) : O for some 01 between 10 and 20. Likewise, f'(cz) = O for some (:2 between 30 and 40. This means we can expand our table: at 010 cl 20 30 62 40 WFWWWWWW This suggests that at) is increasing until m = c1 , and decreasing after, which means there is a local maximum in the interval 10 S a\": S 20 . Moreover, it appears that f(.'L') is decreasing untila': : c2 and increasing after, which means there is a local minimum in the interval 30 S :1: S 40 . Since we can't guarantee the existence of any other critical values in any other intervals, we can't guarantee the existence of extreme values in any other interval