Step on the root!

First, let us calculate all the maxima and minima of $f(x)$ . Looking at its derivative:

$\large \displaystyle f'(x) = 3x^2 - 20x - 11$

Which has roots in:

$\large \displaystyle a = \frac{10 - \sqrt{133}}{3}$ and $\large \displaystyle b = \frac{10 + \sqrt{133}}{3}$ .

Also:

$\large \displaystyle \begin{cases} f'(x) > 0 , x < a \text{ or } x > b \\ f'(x) \leq 0, a \leq x \leq b \end{cases}$

So, the function increases from $- \infty$ until $x = a$ (which is its local maximum), decreases until $x = b$ (which is its local minimum), and increases again from $x =b$ up to $\infty$ .

Also:

$\large \displaystyle f(a) \approx -324.238$

$\large \displaystyle f(b) \approx -97.124$

Both values are smaller than $0$ . So, the only point in which $f(x)$ crosses the $x$ -axis and, therefore, the only root of $f(x)$ is after $x = b$ . Looking up for some integral values of $x$ after $x = b$ :

$\large \displaystyle f(8) = -316$

$\large \displaystyle f(9) = -280$

$\large \displaystyle f(10) = -210$

$\large \displaystyle f(11) = -100$

$\large \displaystyle f(12) = 56$

So, the root has to be between $11$ and $12$ and, therefore, its integral part is $\color{#3D99F6} \boxed{ \large \displaystyle 11}$ .