Value of a and b in a function if its local minimum is at a certain point?

Question

I have a function as written below: $$ g(x) = x^3 + ax^2 + bx $$

It is known to have a local minimum at $ x = -\frac{1}{\sqrt{3}} $ whose value is $ y = -\frac{2\sqrt{3}}{9} $

I have tried using the second derivative test and I got stuck at $ -\frac{6}{\sqrt{3}} + 2a > 0 $ (second derivative at c is > 0)

and $ 1 -\frac{2a}{\sqrt{3}} + b = 0 $ (first derivative equals to 0)

Any ideas?

Answer 1

Also, $$-\frac{1}{3\sqrt3}+\frac{a}{3}-\frac{b}{\sqrt3}=-\frac{2}{3\sqrt3}$$ and solve the system.

I got $$(a,b)=\left(\frac{4}{\sqrt3},\frac{5}{3}\right).$$