Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 50

Use the Intermediate Value Theorem to show that $\tan x = 2x$ has root on the interval $(0.5, 1.4)$

Let $f(x) = \tan x - 2x$
Based from the definition of Intermediate Value Theorem,
There exist a solution c for the function between the interval $(a,b)$ suppose that the function is continuous on that
given interval. So, there exist a number $c$ between 0.5 and 1.4 such that $f(x) = 0$ and that is, $f(c) = 0$.


$\begin{equation} \begin{aligned} f(0.5) & = \tan(0.5) - 2(0.5) = -0.4537\\ f(1.4) & = \tan(1.4) - 2(1.4) = 2.9978 \end{aligned} \end{equation}$


By using Intermediate Value Theorem. We prove that...


So,
$\begin{equation} \begin{aligned} & \text{ if } 0.5 < c < 1.4 \quad \text{ then } \quad f(0.5) < f(c) < f(1)\\ & \text{ if } 0.5 < c < 1.4 \quad \text{ then } \quad -0.4537 < 0 < 2.9978 \end{aligned} \end{equation}$

Therefore,

There exist a such solution $c$ for $\tan x -2x =0$