Question

Find the range of following quadratic expression.
$f\left(x\right)=x^2-2x+3$ $\forall x\in [0,3]$.

Answer 1

Now, $F\left(x\right)=x^2-2x+3$

$=x^2-2x+1+2$

$F\left(x\right)=(x-1{)}^2+2$

we know that $(x-1{)}^2\le 0$

So, $(x-1{)}^2+2\le 2$

hence $(x^2-2x+3)\le 2$

The range of $F\left(x\right)$ is $[2,\infty )$