Answer
$f(2)$ and $f(3)$ have opposite signs,
so f(x) has a real zero between $2$ and $3.$
Work Step by Step
The Intermediate Value Theorem for Polynomial Functions$:$
Let $f$ be a polynomial function with real coefficients.
If $f(a)$ and $f(b)$ have opposite signs,
then there is at least one value of $c$ between $a$ and $b$ for which $f(c)=0$.
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$a=2,\quad f(a)=3(2)^{3}-8(2)^{2}+(2)+2$
$=24-32+2+2=-4\quad$(negative)
$b=3,\quad f(a)=3(3)^{3}-8(3)^{2}+(3)+2$
$=81-72+3+2=14\quad$(positive)
$f(2)$ and $f(3)$ have opposite signs,
so f(x) has a real zero between $2$ and $3.$
