Chapter 4: Applications of Derivatives - Section 4.4 - Concavity and Curve Sketching - Exercises 4.4 - Page 215: 105

$b=-3$

Step 1. Take the derivatives of the given function to get $y'=3x^2+2bx+c$ and $y''=6x+2b$ Step 2. For the function to have an inflection point at $x=1$, let $y''(1)=6+2b=0$, thus $b=-3$. Step 3. Check the signs of $y''=6x-6$ across $x=1$ as: $..(-)..(1)..(+)..$; we see that the function is concave down on $(-\infty,1)$ and concave up on $(1,\infty)$; thus $x=1$ is an inflection point for $b=-3$.