Given the function `g(x)=6x^3-63x^2+216x`, find the first derivative, `g'(x)`.
`g'(x)=`  

Notice that `g'(x)=0` when `x=4`, that is, `g'(4)=0`.

Now, we want to know whether there is a local minimum or local maximum at `x=4`, so we will use the second derivative test.
Find the second derivative, `g''(x)`.
`g''(x)=`  

Evaluate `g''(4)`.
`g''(4)=`

Based on the sign of this number, does this mean the graph of `g(x)` is concave up or concave down at `x=4`?
[Answer either up or down -- watch your spelling!!]
At `x=4` the graph of `g(x)` is concave

Based on the concavity of `g(x)` at `x=4`, does this mean that there is a local minimum or local maximum at `x=4`?
[Answer either minimum or maximum -- watch your spelling!!]
At `x=4` there is a local