Answer
False.
Work Step by Step
There is a condition missing if we want to apply the Intermediate Value Theorem,
f must also be continuous for this to be true.
Construct a piecewise function, such as
$f(x)=\left\{\begin{array}{lll}
1, & & x < 2\\
-1 & & x \geq 2
\end{array}\right.$
which fits the conditions of the statement,
but there is no c in the interval [1,3] for which f(c)=0.
