The equation

ex $=32$x

is equivalent to the equation

f$($x$)=$ ex $3+2$x $=0.$

f$($x$)$

is continuous on the interval

$[0,1],$

f$(0)=$

$2$

$,$

and

f$(1)=$

$1\mathrm{.}718$

$.$

Since

$---$Select$---$ f$(0)$ f$(1)<0<---$Select$---$ f$(0)$ f$(1),$

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

ex $=32$x$,$

in the interval

$(0,1).$