The Best Max of Two Absolutes

Let $f(x) = |x + a| + |x^2 + b|$ for $x \in [0, 1]$, and let $M(a, b)$ denote the maximum of $f(x)$ on $[0, 1]$. If the minimum possible value of $M(a, b)$ is $1$, find all possible values of $a$.