Context: The function $f(x) = (x - a)(x - b)$ has integer constants $a$ and $b$, with $f(14) > 0$, $f(17) < 0$, and $f(20) > 0$. We need the lowest possible value of $a + b$.

Answer: 33

Why: This upward-opening parabola is negative strictly between its roots and positive outside them. $f(17) < 0$ forces $17$ to lie strictly between the roots, while $f(14) > 0$ and $f(20) > 0$ force $14$ and $20$ to lie strictly outside. Labeling the roots so that $a < b$, this means $14 < a < 17 < b < 20$. The integer possibilities are $a \in \{15, 16\}$ and $b \in \{18, 19\}$, so $a + b$ can be $33$, $34$, or $35$. The lowest possible value is $15 + 18 = 33$.

Verification: With roots $15$ and $18$: $f(14) = (-1)(-4) = 4 > 0$, $f(17) = (2)(-1) = -2 < 0$, and $f(20) = (5)(2) = 10 > 0$; all three conditions hold.

Common slips: Allowing $a = 14$ or $b = 20$; the strict inequalities exclude the endpoints, because making $14$ or $20$ a root would force $f(14) = 0$ or $f(20) = 0$ rather than a positive value.