Functions Ans 18

Question 18 - 31 January - Shift 2

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 :

(1) $\frac{3}{4}$

(2) $\frac{3}{2}$

(3) $\frac{1}{4}$

(4) $\frac{5}{4}$

Show Answer

Answer: (1)

Solution:

Formula: Properties of Greatest Integer Function, Operations on functions, Maximum and Minimum value

$ f(x)=|x^{2}-x+1|+\lfloor x^{2}-x+1\rfloor ; x \in\lfloor-1,2\rfloor $

Let $g(x)=x^{2}-x+1$

$=(x-\frac{1}{2})^{2}+\frac{3}{4}$

$\because|x^{2}-x+1|$ and $[x^{2}-x+2]$

Both have minimum value at $x=1 / 2$

$\Rightarrow$ Minimum $f(x)=\frac{3}{4}+0=\frac{3}{4}$