Answer
$f(x)$ is increasing on $(−∞, ∞)$ if $ b \gt a^2/3$
Work Step by Step
Given $$f(x)=x^{3}+a x^{2}+b x+c$$
Since
\begin{align*}
f'(x)& = 3x^2+2ax +b \\
&= 3\left[x^2+\frac{2a}{3}x\right]+b\\
&= 3\left( x+\frac{a}{3}\right)^2+\left(b- \frac{a^2}{3}\right)
\end{align*}
Then $f'(x)\gt0$ when $b\gt \dfrac{a^2}{3}$. Hence, $f(x)$ is increasing on $(−∞, ∞)$ if $ b \gt a^2/3$.
