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Q.
The absolute minimum value, of the function $f(x)=\left|x^2-x+1\right|+\left[x^2-x+1\right]$, where $[t]$ denotes the greatest integer function, in the interval $[-1,2]$, is:
$f(x)=\left|x^2-x+1\right|+\left[x^2-x+1\right] ; x \in[-1,2]$
Let $g( x )= x ^2- x +1$
$=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}$
$\because\left| x ^2- x +1\right| \text { and }\left[ x ^2- x +2\right]$
Both have minimum value at $x=1 / 2$
$ \Rightarrow $ Minimum $ f ( x )=\frac{3}{4}+0$
$=\frac{3}{4}$