Question

Find the intervals in which the function $f$ given by $f\left(x\right)=x^2-4x+6$ is
(a) increasing (b) decreasing

Answer 1

Given:
$f\left(x\right)=x^2-4x+6$

Differentiating w.r.t. $x$, we get

$f^{\prime }\left(x\right)=2x-4=2(x-2)$

Now,

$f^{\prime }\left(x\right)=0$ at $x=2$

$f^{\prime }\left(x\right)<0,\text{when}xϵ(-\infty ,2)$

$f^{\prime }\left(x\right)>0,\text{when}xϵ(2,\infty )$

Hence,

$f$ is increasing in interval $(2,\infty )$

$f$ is decreasing in interval $(-\infty ,2)$