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

$f (x) = | x^{2} - x + 1 | + [x^{2} - x + 1]$
$| x^{2} - x + 1 |$ attains min. $x = \frac{1}{2}$
$f (\frac{1}{2}) = | \frac{3}{4} | + [3 / 4] = \frac{3}{4} + 0 = \frac{3}{4}$

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