Answer
Check work for explanation.
Work Step by Step
Since $f(x)$ is a polynomial, it is continuous for all values of $x.$
$f(1)=\frac{1}{12}(1)^4-(1)^3+4=\dfrac{37}{12}\to f(1)\gt0.$
$f(2)=\frac{1}{12}(2)^4-(2)^3+4=-\dfrac{8}{3}\to f(2)\lt0.$
Since $f(x)$ is continuous over the interval $[1, 2]$ and the sign of $f(x)$ changes over the interval $[1, 2]$, then the Intermediate Value Theorem guarantees at least one root in the interval.
