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4-2.Quadratic Equations and Inequations
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Let $f(x)={{x}^{2}}-x+k-2,k\in R$ then the complete set of values of $k$ for which $y=\left| f\left( \left| x \right| \right) \right|$ is non-derivable at $5$ distinict points is
A
$(1,4)$
B
$\left( 0,\frac{9}{4} \right)$
C
$\left( -\infty ,2 \right)$
D
$\left( 2,\frac{9}{4} \right)$
Solution
Both roots of the equation $x^{2}-x+k-2=0$ must be positive and distinct.
$\therefore D>0 \Rightarrow 1-4(\mathrm{k}-2)>0 \Rightarrow \mathrm{k}<\frac{9}{4}$
$\alpha \beta>0 \Rightarrow \mathrm{k}-2>0 \Rightarrow \mathrm{k}>2$
$\alpha+\beta>0 \Rightarrow$ true
$\therefore k \in\left(2, \frac{9}{4}\right)$
Standard 11
Mathematics
