This problem comes from Edward J. Gaughan's Introduction to Analysis.
(4.3.19) Show that has precisely one root on the closed interval
Let . Since each component of is differentiable and therefore continuous on the interval. Then by the Intermediate Value Theorem, since and , such that .
Since , if , then , is monotonic increasing and therefore , .
The derivative of at a point is the limit of as . Is there a way to show that this limit is strictly greater than zero? Is it sufficient to find the limit of at and then say that since is monotonic increasing, that this would be a lower bound for the derivative of ?
EDIT: I should mention that I want to show c is unique. So I'd like to say the limit is strictly greater than zero so that I can apply Rolle's Theorem.