Calculus, derivatives (curve sketching 2)

[physics604]

1. If the function f(x)=x³+a²+bx has the local minimum value at $\frac{-2}{9}$$\sqrt{3}$, what are the values of
and a and b?

Homework Equations

$$f'(x)=0$$

The Attempt at a Solution

I automatically took the derivative, getting $$f'(x)=3x^2+2ax+b$$ However, I have no idea where to go from here because I only know one root ($\frac{-2}{9}$$\sqrt{3}$) and not the other. Can someone give me a hint?

 

[Simon Bridge]

Your derivative does not follow from the given function.
$$f(x)=x^3+a^2+bx \implies f'(x)=3x^2+b$$ Your derivative is of $$f(x)=x^3+ax^2+bx+c$$ ... which is correct?

(I'm kinda leaning towards the second one with c=0 but I'd like to be sure.)

You are supposed to use your understanding of cubic equations to help you, not just algebra.
I suspect that has been the problem in both your questions I've seen so far.

 

[Mentallic]

Do you know any relationships between the inflexion point of a cubic with that of its turning points? If not, use a graphing calculator to sketch a few cubics that have distinct local min and max points, and see if you can notice anything between those and the inflexion point. Maybe try finding the inflexion point in each example to make it more obvious.

 

[Simon Bridge]

Lynchpin: roots of f(x) and f'(x).
(Assuming my suspicion is correct.)