The statement is false.
Work Step by Step
The statement follows the result of the Intermediate Value Theorem, which states that let $N$ be any number between $f(a)$ and $f(b)$ and $f(a)\ne f(b)$, then there exists a number $c$ in $(a,b)$ such that $f(c)=N$. However, the Intermediate Value Theorem can only be applied when the function $f(x)$ is continuous on interval $[a,b]$. In this case, we are not sure whether function $f(x)$ is continuous on interval $[1,3]$ or not. It only proves that $f(1)\ne f(3)$, since $f(1)\gt0$ and $f(3)\lt0$ Therefore, the statement is inadequate and false as a result.