Question:
$f(x)=(x-1)^{2}+2$ on $R$
Solution:
Given: $f(x)=-(x-1)^{2}+2$
Now,
$(x-1)^{2} \geq 0$ for all $x \in \mathbf{R}$
$\Rightarrow f(x)=-(x-1)^{2}+2 \leq 2$ for all $x \in \mathbf{R}$
The maximum value of $f(x)$ is attained when $(x-1)=0$.
$(x-1)=0$
$\Rightarrow x=1$
Therefore, the maximum value of $f(x)=2$
Since $f(x)$ can be reduced, the minimum value does not exist, which is evident in the graph also.
Hence, function $f$ does not have a minimum value.