Example 5 suppose that f(0) = −7 and f '(x) ≤ 8 for all values of x. how large can f(3) possibly be? solution we are given that f is differentiable (and therefore continuous) everywhere. in particular, we can apply the mean value theorem on the interval [0, 3] . there exists a number c such that f(3) − f(0) = f '(c) 3 correct: your answer is correct. − 0 so f(3) = f(0) + 3 correct: your answer is correct. f '(c) = −7 + 3 correct: your answer is correct. f '(c). we are given that f '(x) ≤ 8 for all x, so in particular we know that f '(c) ≤ 8 correct: your answer is correct. multiplying both sides of this inequality by 3, we have 3f '(c) ≤ 24 correct: your answer is correct. , so f(3) = −7 + incorrect: your answer is incorrect. f '(c) ≤ −7 + incorrect: your answer is incorrect. = incorrect: your answer is incorrect. the largest possible value for f(3) is .

In a study of the relationship between socio-economic class and unethical behavior, 129 university of california undergraduates at berkeley were asked to identify themselves as having low or high social-class by comparing themselves to others with the most (least) money, most (least) education, and most (least) respected jobs. they were also presented with a jar of individually wrapped candies and informed that the candies were for children in a nearby laboratory, but that they could take some if they wanted. after completing some unrelated tasks, participants reported the number of candies they had taken. it was found that those who were identiď¬ed as upper-class took more candy than others. in this study, identify: (a) the cases, (b) the variables and their types, (c) the main research question, (d) identify the population of interest and the sample in this study, and (e) comment on whether or not the results of the study can be generalized to the population, and if the ď¬ndings of the study can be used to establish causal relationships.