Consider the following statement of the intermediate value theorem for derivatives:
Assume is differentiable on an open interval . Let be two points in . Then, the derivative takes every value between and somewhere in .
- Define a function
Prove that takes every value between and in the interval . Then, use the mean-value theorem for derivatives to show takes all values between and somewhere in the interval .
- Define a function
Show that the derivative takes on all values between and in the interval . Conclude that the statement of the intermediate-value theorem is true.
- Proof. First, since is differentiable everywhere on the interval , we know is continuous on and differentiable on . Thus, if and then is continuous at since it is the quotient of continuous functions and the denominator is nonzero. If then
hence, is continuous at as well. Therefore, is continuous on the closed interval . So, by the intermediate value theorem for continuous functions we know takes on every value between and somewhere on the interval . Since this means takes on every value between and somewhere on the interval .
By the mean-value theorem for derivatives, we then know there exists some such thatfor some . Since , we then conclude there is some such that for any . Since takes on every value between and , so does .
- Proof. This is very similar to part (a). By the same argument we have the function is continuous on ; thus, takes on every value between and by the intermediate value theorem for continuous functions. Then, by the mean value theorem, we know there exists a such that
Thus, takes on every value between and . Since ; takes on every value between and