$f(x) = x^{3} + ax^{2} + bx$ has local extrema at $x = 1$ and $x = 3$. What are $a$ and $b$?

Explanation

1. 01

   $f'(x) = 3x^{2} + 2ax + b$ must be 0 at both points: $3 + 2a + b = 0$ and $27 + 6a + b = 0$.

2. 02

   Subtracting: $24 + 4a = 0$, so $a = -6$ and $b = 9$. Check: $f'(x) = 3(x - 1)(x - 3)$.