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The absolute minimum value, of the function $f (x) = |x^{2} - x + 1| + [x^{2} - x + 1] where [t]$ denotes the greatest integer function, in the interval [-1,2] is:

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$\frac{3}{4}$

$\frac{5}{4}$

$\frac{3}{2}$

$\frac{1}{4}$

answer is D.

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Detailed Solution

$f (x) = |x^{2} - x + 1| + [x^{2} - x + 1]$
$([.] is GIF \leq x) in [- 1, 2]$
$|x^{2} - x + 1| attains this minimum x = \frac{1}{2}$
$f (\frac{1}{2}) = |\frac{3}{4}| + [\frac{3}{4}] = \frac{3}{4}$

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