Given the function $\displaystyle {g{{\left({x}\right)}}}={6}{x}^{{3}}+{63}{x}^{{2}}+{216}{x}$, find the first derivative, $\displaystyle {g}'{\left({x}\right)}$.
$\displaystyle {g}'{\left({x}\right)}=$Preview Question 6 Part 1 of 5  

Notice that $\displaystyle {g}'{\left({x}\right)}={0}$ when $\displaystyle {x}=-{4}$, that is, $\displaystyle {g}'{\left(-{4}\right)}={0}$.

Now, we want to know whether there is a local minimum or local maximum at $\displaystyle {x}=-{4}$, so we will use the second derivative test.
Find the second derivative, g"$\displaystyle {\left({x}\right)}$.
g"$\displaystyle {\left({x}\right)}=$Preview Question 6 Part 2 of 5  

Evaluate g"$\displaystyle {\left(-{4}\right)}$.
g"$\displaystyle {\left(-{4}\right)}=$

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

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