The first derivative of the function $f$ is given by $f'(x)=\dfrac{3\cos^2 x}{x}-\dfrac{1}{6}$. How many critical values does $f$ have on the open interval $(0,10)$ ?

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To find the critical values, set the derivative equal to 0. In the graphing calculator, we can enter the equation and find the $x$-intercepts.

As shown below, the graph of the derivative has 6 $x$-intercepts on the interval $(0,10)$.

Critical values also occur where $f'(x)$ is undefined. This occurs at $x=0$, but our interval is $(0,10)$ exclusive of $0$.