If the solve the problem


$f(x)=(x-1)^{2}+2$ on $R$


Given: $f(x)=-(x-1)^{2}+2$


$(x-1)^{2} \geq 0$ 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 $(x-1)=0$.


$\Rightarrow x=1$

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.

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