This problem comes from Edward J. Gaughan's Introduction to Analysis.

*The Problem*
(4.3.19) Show that

has precisely one root on the closed interval

*Attempted Solution*

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 issue*

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

?