Cubic function

[starsnstripes900]

Find the cubic function g(x)=ax^3+bx^2+cx+d that has a local max value 3 at -2 and a local min value 0 at 1. I need all the help I can get! Thank you! and show all work please!

 

[soroban]

Hello, starsnstripes900!

\(\displaystyle \text{Find the cubic function }g(x)\:=\:ax^3+bx^2+cx+d\,\text{ that has a local max at }(-2,3)\text{ and a local min at }(1,0).\)

\(\displaystyle \text{We are told that: }\:g(\text{-}2) \,=\,3\)
. . \(\displaystyle \text{We have: }\:\text{-}8a + 4b - 2c + d \:=\:3\)

\(\displaystyle \text{We are told that: }\:g(1) \,=\,0\)
. . \(\displaystyle \text{We have: }\:a + b + c + d \:=\:0\)


\(\displaystyle \text{We have: }\:g'(x) \:=\:3ax^2 + 2bx + c\)

\(\displaystyle \text{We are told that: }\:g'(-2) \,=\,0.\)
. . \(\displaystyle \text{We have: }\:12a - 4b + c \:=\:0\)

\(\displaystyle \text{We are told that: }\:g'(1) \,=\,0\)
. . \(\displaystyle \text{We have: }\:3a + 2b + c \:=\:0\)

\(\displaystyle \text{Solve the system of equations: }\;\begin{array}{ccccc} \text{-}8a + 4b - 2c + d &=& 3 \\ a + \;b + \;c + \;d &=& 0 \\ 12a - 4b + c \qquad &=& 0 \\ 3a + 2b + c \qquad &=& 0 \end{array}\)

 

[starsnstripes900]

is all the work shown here? it seems so simple yet my professor told us that he did not put this question on the test because it is so involved.