Question

Find the maximum and minimum values, if any, without using derivatives of the following function.
$f\left(x\right)=-(x-1{)}^2+2$ on $R$.

Answer 1

Given,

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

we have, $-(x-1{)}^2\le 0$ for all $x\in R$

$\Rightarrow -(x-1{)}^2+2\le 0+2$ for all $x\in R$

$\therefore f\left(x\right)\le 2$ for all $x\in R$

$\therefore x=2$ is the maximum value.

The function doesn't attain the minimum value at any point in its domain.