In the last section, we discussed how to differentiate implicitly. In this section, we are going to discuss some common theorems from differential calculus including the sandwich theorem, maximum-minimum theorem, intermediate-value theorem, and the mean value theorem.
Let's start with the sandwich theorem. This theorem also goes by the name of the pinching theorem. The sandwich theorem starts with some assumptions and those are must be greater than 0, and for all x in the form 0 < |x - c| < , g(x) f(x) h(x). Furthermore, and . What the assumptions are saying is we are considering all x such that x < c + and x > c - , and for those x values, g(x) f(x) h(x). Finally if the two above listed limits hold then, . The result says that the limit of f(x) goes to L if the limits of g(x) and h(x) also go to L.
The intermediate-value theorem also has some assumptions. Those are f(x) is continuous on [a,c], and B is a number between f(a) and f(c). If those two criteria are true then there also exists a number b such that f(b)=B. What the theorem is saying is that f takes on all values between f(a) and f(c), and for those values there exists a number in the domain of f called b which is between a and c. We can make this number b take on all the any value in the range of f which is between f(a) and f(c). Well, it may not take on any value in the range just a subset of it because f may be defined elsewhere other than the interval [a,c].
The maximum-minumum theorem is a lot simpler. The assumptions are also fewer, and that is f is continuous on [a,b]. If so, f will have a maximum and minimum on [a,b]. Not much to this theorem.
The mean value theorem's assumptions are f is differentiable on (a,c) and continuous on [a,c]. If so, then there is a number b such that . You should recognize the fraction as the slope of the line between (a,f(a)) and (c,f(c)). The theorem is saying the derivative of f will take on the value of the slope of the line between the points (a,f(a)) and (c,f(c)).