Mathematics

The key is recognizing that concavity is determined by the second derivative, and an inflection point at $x=0$ means $F''(0)=0$ with the second derivative changing sign there. Since $F''(x)=6ax^2+2b$, you need $F''(0)=2b=0$, so $b=0$ is necessary. For sufficiency, check that $b=0$ alone forces the concavity to flip at $x=0$ regardless of $a$'s sign—it does, because then $F''(x)=6ax^2$ has a single zero at the origin with the required sign change when $a\neq 0$.

The key is recognizing that concavity is determined by the second derivative, and an inflection point at $x=0$ means $F''(0)=0$ with the second derivative changing sign there. Since $F''(x)=6ax^2+2b$, you need $F''(0)=2b=0$, so $b=0$ is necessary. For sufficiency, check that $b=0$ alone forces the concavity to flip at $x=0$ regardless of $a$'s sign—it does, because then $F''(x)=6ax^2$ has a single zero at the origin with the required sign change when $a\neq 0$.