1. ## derivatives question story

Given the function ${g{{\left({x}\right)}}}={8}{{x}}^{{3}}-{96}{{x}}^{{2}}+{360}{x}$, find the first derivative, ${g{'}}{\left({x}\right)}$.
${g{'}}{\left({x}\right)}=$

Notice that ${g{'}}{\left({x}\right)}={0}$ when ${x}={5}$, that is, ${g{'}}{\left({5}\right)}={0}$.

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

Evaluate g"${\left({5}\right)}$.
g"${\left({5}\right)}=$

Based on the sign of this number, does this mean the graph of ${g{{\left({x}\right)}}}$ is concave up or concave down at ${x}={5}$?
At ${x}={5}$ the graph of ${g{{\left({x}\right)}}}$ is concave

Based on the concavity of ${g{{\left({x}\right)}}}$ at ${x}={5}$, does this mean that there is a local minimum or local maximum at ${x}={5}$?
At ${x}={5}$ there is a local Box 1: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c
Be sure your variables match those in the question
Box 2: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c
Be sure your variables match those in the question
Box 3: Enter your answer as a whole or decimal number. Examples: 3, -4, 5.5
Enter DNE for Does Not Exist, oo for Infinity
Box 4: Enter your answer as letters. Examples: A B C, linear, a cat
Box 5: Enter your answer as letters. Examples: A B C, linear, a cat

2. ## Re: derivatives question story

what have you done with this problem?