Answer
a) Local maximum at $x = 2$
b) Inconclusive
Work Step by Step
a) By the Second Derivative Test if $f'(2) = 0$ and $f''(2) = -5 \lt 0$, $f$ has a local maximum at $x = 2$.
b) If $f'(6) = 0$, we know that $f$ has a horizontal tangent at $x = 6$.
Knowing that $f''(x) = 0$ does not provide any additional information since the Second Derivative Test fails (it is inconclusive). At $x=6$ $f$ can either have a minimum, a maximum or neither.