Zeros Inside the Unit Interval

Let $f(x) = x^3 + bx + c$, and suppose the tangent to $y = f(x)$ at $x = \dfrac12$ is perpendicular to the $y$-axis.

1. Find $b$. 2. If $f$ has a zero of absolute value at most $1$, prove that all zeros of $f$ have absolute value at most $1$.