Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 189: 51

$f(x) $ is decreasing on $\left(-\infty,-\frac{b}{2}\right)$ and increasing on $\left(-\frac{b}{2}, \infty\right)$

Given $$f(x)=x^{2}+b x+c$$ Since $$ f'(x)= 2x+b $$ Then $f'(x)=0$ for $x=\dfrac{-b}{2}$. Since $f'(x)\lt 0$ for $x\lt -b/2$ and $f'(x)\gt 0$ for $x>\dfrac{-b}{2}$, then $f(x) $ is decreasing on $\left(-\infty,-\frac{b}{2}\right)$ and increasing on $\left(-\frac{b}{2}, \infty\right)$.