Solve for .
By mere inspection, we have .
Visually, it appears that is the only solution (see Fig.1 or Fig. 2)
To show that is the only solution of analytically, let
is a solution of means
Suppose there is a solution
Since is an function continuous and differentiable on ,
by Lagrange’s Mean-Value Theorem (see “A Sprint to FTC“), there such that
From (5), we have
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 .