Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 22 Marks
Question
Prove that the function $f$ given by $f(x)=x^2-x+1$ is neither strictly increasing nor strictly decreasing on $(-1,1)$.
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Answer
Given: $f(x)=x^2-x+1 f ( x )= x ^2- x +1$
$\Rightarrow f^{\prime}(x)=2 x-1$$f ( x )$ is strictly increasing if $f ( x )<0$
$ \Rightarrow 2 x-1>0$
$\Rightarrow x>\frac{1}{2} $
i.e., increasing on the interval $\left(\frac{1}{2}, 1\right)$
$f(x)$ is strictly decreasing if $f(x)<0$
$ \Rightarrow 2 x-1<0$
$\Rightarrow x<\frac{1}{2} $
i.e., decreasing on the interval $\left(-1, \frac{1}{2}\right)$
hence, f(x) is neither strictly increasing nor decreasing on the interval $(-1, 1).$
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