Solution (a) The first derivative of \(f\) is \[ f^\prime (x)=3x^{2}-12x+9=3( x-1) (x-3) \]
So, \(1\) and \(3\) are critical numbers of \(f.\) Now,
So \(f\) is increasing on \(( -\infty ,1)\) and on \((3,\infty)\); \(f\) is decreasing on \((1,3)\).
At \(1,\) \(f\) has a local maximum, and at \(3\), \(f\) has a local minimum. The local maximum value is \(f( 1) = 34\); the local minimum value is \(f(3) =30\).
(b) To determine concavity, we use the second derivative: \[ f^{\prime \prime} (x)=6x-12= 6(x-2) \]
Now, we solve the inequalities \(f^{\prime \prime} (x) <0\) and \(f^{\prime \prime} (x) >0\) and use the Test for Concavity.