How Much Bend Before the Bowl Breaks

A team is designing a loss function of one parameter $x$. They have narrowed it to this family, where $b$ is a constant they are free to choose:

$f(x) = x^4 + b\,x^3 + 15x^2 + 2x + 9$

They want the loss to be convex over the whole real line — a single bowl, no side dips — so that any flat point their search reaches is guaranteed to be the global minimum rather than a trap.

For a twice-differentiable function, convexity is the condition

$f''(x) \ge 0 \quad \text{for } \textbf{every } x$

not merely at the point they happen to start from. A large $b$ tilts the cubic term hard enough to punch a second valley into the curve; a small enough $b$ leaves the bowl intact.

What is the largest value of $b$ for which $f$ is convex everywhere?

Round your answer to 2 decimal places.