If the solve the problem

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.

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