Question

Find the maximum and minimum values, if any, of the following function given by,

$f\left(x\right)=-(x-1{)}^2+10$

Answer 1

Given, function is $f\left(x\right)=-(x-1{)}^2+10$
It can be observed that $(x-1{)}^2\ge 0$ for all $xϵR$
$\Rightarrow -(x-1{)}^2\le 0$ for every $xϵR$
Therefore, $f\left(x\right)=-(x-1{)}^2+10\le 10$ for every $xϵR$
The maximum value of f is attained when (x-1)=0
i.e., $(x-1)=0\Rightarrow x=1\therefore \text{Maximum value of}f=f\left(1\right)=-(1-1{)}^2+10=10$
For any value of x, $f\left(x\right)\le 10$, hence function f does not have a particular minimum value.