Solve for .

By mere inspection, we have .

Visually, it appears that is the only solution (see Fig.1 or Fig. 2)

Fig. 1

Fig. 2

To show that is the only solution of *analytically, *let

is a solution of means

Suppose there is a solution

then,

Since is an function continuous and differentiable on ,

by Lagrange’s Mean-Value Theorem (see “A Sprint to FTC“), there such that

.

We know

.

From (5), we have

.

i.e.,

.

This is not possible since .

A simpler alternative without direct applying Lagrange’s Mean-Value Theorem is:

is a strictly decreasing function.

Since and .

Therefore, is the only solution of . i.e.,

is the only solution of .