The absolute minimum value, of the function f(x)=|x2-x+1|+[x2-x+1], where [t] denotes the greatest integer function, in the interval [-1,2], is:

Question

The absolute minimum value, of the function f(x)=|x2-x+1|+[x2-x+1], where [t] denotes the greatest integer function, in the interval [-1,2], is: (A) (3/2) (B) (1/4) (C) (5/4) (D) (3/4)

Tardigrade Admin · Accepted Answer

f(x)=|x2-x+1|+[x2-x+1] ; x ∈[-1,2] Let g( x )= x 2- x +1 =(x-(1/2))2+(3/4) ∵| x 2- x +1| text and [ x 2- x +2] Both have minimum value at x=1 / 2 ⇒ Minimum f ( x )=(3/4)+0 =(3/4)

Q. 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:

$\frac{3}{2}$

$\frac{1}{4}$

$\frac{5}{4}$

$\frac{3}{4}$

$f (x) = ∣ ∣ x^{2} - x + 1 ∣ ∣ + [x^{2} - x + 1]; x \in [- 1, 2]$
Let $g (x) = x^{2} - x + 1$
$= (x - \frac{1}{2})^{2} + \frac{3}{4}$
$∵ ∣ ∣ x^{2} - x + 1 ∣ ∣ and [x^{2} - x + 2]$
Both have minimum value at $x = 1/2$
$\Rightarrow$ Minimum $f (x) = \frac{3}{4} + 0$
$= \frac{3}{4}$

Q. The absolute minimum value, of the function f(x)=∣∣​x2−x+1∣∣​+[x2−x+1], where [t] denotes the greatest integer function, in the interval [−1,2], is:

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