Given the function `g(x)=6x^3+63x^2+180x`, find the first derivative, `g'(x)`.
`g'(x)=`
Notice that `g'(x)=0` when `x=-2`, that is, `g'(-2)=0`.
Now, we want to know whether there is a local minimum or local maximum at `x=-2`, so we will use the second derivative test.
Find the second derivative, `g''(x)`.
`g''(x)=`
Evaluate `g''(-2)`.
`g''(-2)=`
Based on the sign of this number, does this mean the graph of `g(x)` is concave up or concave down at `x=-2`?
[Answer either up or down -- watch your spelling!!]
At `x=-2` the graph of `g(x)` is concave
Based on the concavity of `g(x)` at `x=-2`, does this mean that there is a local minimum or local maximum at `x=-2`?
[Answer either minimum or maximum -- watch your spelling!!]
At `x=-2` there is a local
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