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 \(0\).