How can I find a cubic function with specific local maximum and minimum values?

[ryan.1015]

Homework Statement

find a cubic function g(x)=ax^3 +bx^2+cx +d that has a local maximum value of 3 at -7 and a local minimum value 0f -9 at 12.

Homework Equations

The Attempt at a Solution

I know the derivative should equal zero for a max or min to occure. So i got f '(x)=(x+7)(x-12). then i got F '(x)=x^2-5x-84 and plugged that into the original equation. I got A=1/3 b=5/2 and c=-84. I'm not sure how to fet D, or if i did this first part right

 

[rock.freak667]

Basically you know that g(3)=-7 and g(-9)=12. And also that g'(3)=g'(9)=0. Solve now.


EDIT: It should be g(7)=-3 and g(12)=9, not the other way around

 

[Mark44]

You switch function names, with the original starting out as g(x).

g'(x) doesn't have to be exactly (x + 7)(x - 12). It could be a constant multiple of this expression, namely g'(x) = A(x + 7)(x - 12) = A(x^2 - 5x - 84). Also, you can calculate g'(x) from the original equation for g, and compare this to the one above.

You know that g(-7) = 3 and that g(12) = -9.

What about the second derivative? You can calculate g''(x) from the equation above, as well as from the original equation. What do you know about the value of the second derivative at a local maximum? At a local minimum?

You should be more careful with your notation. You have referred to the original function as g, f, and F. Also, the coefficients of the equation for g(x) involved a, b, c, and d, not A.