Answer
$f(0)=1$
$f(1)=0.46$
There is a root of the equation $cos~ x − x= 0,$ or
$cos~ x=x,$ in the interval $(0, 1)$.
Work Step by Step
Given:
$f(x) = cos~ x − x$.
The given function $f(x) = cos~ x − x$ is continuous on the
interval [0, 1],
$f(0) =cos~(0) − 0 $
$= 1 − 0 $
$f(0)=1$,
and
$f(1) =cos~1 -1 $
$f(1)≈ −0.46.$
Since $-0.46<0 <1 ,$ there is a number $c$
in $(0, 1)$ such that $f(c)=0$ by the Intermediate Value Theorem. Thus, there is a root of the equation $cos~ x − x= 0,$ or
$cos~ x=x,$ in the interval $(0, 1)$.
