Answer
Check work for explanation.
Work Step by Step
Since $f(x)$ is a polynomial, it is continuous for all values of $x.$
$f(0)=(0)^3+5(0)-3=-3\to f(0)\lt0.$
$f(1)=(1)^3+5(1)-3=3\to f(1)\gt0$
Since $f(x)$ is continuous over the interval $[0, 1]$ and the sign of $f(x)$ changes over the interval $[0, 1]$, then the Intermediate Value Theorem guarantees at least one root in the interval.
