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 :
Solution
$\mathrm{f}(\mathrm{x})=\left|\mathrm{x}^{2}-\mathrm{x}+1\right|+\left[\mathrm{x}^{2}-\mathrm{x}+1\right] ; \mathrm{x} \in[-1,2]$
<br/><br/>Let $g(x)=x^{2}-x+1$
<br/><br/>$=\left(x-\frac{1}{2}\right)^{2}+\frac{3}{4}$
<br/><br/>$$
\because\left|\mathrm{x}^{2}-\mathrm{x}+1\right| \text { and }\left[\mathrm{x}^{2}-\mathrm{x}+1\right]
$$
<br/><br/>Both have minimum value at $\mathrm{x}=1 / 2$
<br/><br/>$\Rightarrow$ Minimum $\mathrm{f}(\mathrm{x})=\frac{3}{4}+0$
<br/><br/>$=\frac{3}{4}$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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