The Mean Value Theorem and Inequalities The mean value theorem tells us that if f and f � are continuous on [a, b] then: f (b) − f (a) = f � (c) b−a for some value c between a and b. Since f � is continuous, f � (c) must lie between the minimum and maximum values of f � (x) on [a, b]. In other words: min f � (x) ≤ a≤x≤b f (b) − f (a) = f � (c) ≤ max f � (x). b−a a≤x≤b This is the form that the mean value theorem takes when it is used in problem solving (as opposed to mathematical proofs), and this is the form that you will need to know for the test. In practice, you may even forget the mean value theorem and remember only these three inequalities: • If f � (c) > 0 then f (b) > f (a). • If f � (c) < 0 then f (b) < f (a). • If f � (c) = 0 then f (b) = f (a). These can be used to prove mathematical inequalities. The following exam ples compare the function ex to its linear and quadratic approximations and are the ﬁrst steps toward a deeper understanding of the function. Example: Show that ex > 1 + x for x > 0. To prove this, we’ll instead show that f (x) = ex − (1 + x) is always positive. We know that f (0) = e0 − (1 + 0) = 0 and f � (x) = ex − 1. When x is positive, f � (x) is positive because ex > 1. We know that if f � (x) > 0 on an interval then f (x) is increasing on that interval, so we can conclude that f (x) > f (0) for x > 0. In other words, ex − (1 + x) > 0 ⇐⇒ ex > 1 + x. 2 Example: Show that ex > 1 + x + x2 for x > 0. 2 The value of 1 + x + x2 is slightly greater than that of 1 + x, but it turns 2 out that it’s still less than the value of ex . We let g(x) = ex − (1 + x + x2 ) and do the same thing we did before: g(0) = 1 − (1) = 0 g � (x) = ex − (1 + x) We know g � (x) > 0 because we proved f (x) > 0 in the above example. Since g � (x) is positive, g is increasing for x > 0, so g(x) > g(0) when x > 0, so 2 2 ex − (1 + x + x2 ) > 0 and ex > (1 + x + x2 ). x2 x3 We can keep on going: ex > 1 + x + + for x > 0. Eventually, it turns 2 3! out that x2 x3 ex = 1 + x + + + · · · (an inﬁnite sum) 2 3! We will be discussing this when we get to Taylor series near the end of the course.