Question

Find all critical values of the function g(x)=5/(x²-x+6)

Answer 1

$g(x)=\frac{5}{x^2-x+6}\\\\D:x^2-x+6\neq0\\\\\Delta=(-1)^2-4\cdot1\cdot6=1-24=-23 < 0\\\\x\in\mathbb{R}\\\\vertex\ of\ y=x^2-x+6\\\\p=\frac{-(-1)}{2\cdot1}=\frac{1}{2}\\\\q=\frac{-(-23)}{4\cdot1}=\frac{23}{4}\\\\\lim\limits_{x\to\frac{1}{2}}\frac{5}{x^2-x+6}=\frac{5}{\frac{23}{4}}=\frac{5\cdot4}{23}=\frac{20}{23}$

$\lim\limits_{x\to\pm\infty}\frac{5}{x^2-x+6}=0$