Find the maximum and the minimum values, if any, without using derivatives of the following functions: f (x) = –(x – 1)^2 + 2 on R

Given $f(x)=-(x-1)^{2}+2$
It can be observed that $(x-1)^{2} \geq 0$ for every $x \in R$
Therefore, $f(x)=-(x-1)^{2}+2 \leq 2$ for every $x \in R$
The maximum value of $f$ is attained when $(x-1)=0$
$
(x-1)=0, x=1
$
Since, Maximum value of $f=f(1)=-(1-1)^{2}+2=2$
Hence, function $f$ does not have minimum value.