Let f(x) = x^{2} - x + k - 2,where k in R. Fi | vedclass

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(D) The function $y = |f(|x|)|$ is non-derivable at $5$ distinct points if the quadratic equation $f(x) = x^{2} - x + k - 2 = 0$ has two distinct posi

Vedclass · Accepted Answer

$(2, \frac{9}{4})$

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(D) The function $y = |f(|x|)|$ is non-derivable at $5$ distinct points if the quadratic equation $f(x) = x^{2} - x + k - 2 = 0$ has two distinct positive roots.First,for the roots to be real and distinct,the discriminant $D$ must be greater than $0$:$D = (-1)^{2} - 4(1)(k - 2) > 0$$1 - 4k + 8 > 0$$9 - 4k > 0 \Rightarrow k Second,for both roots to be positive,the product of the roots must be positive and the sum of the roots must be positive:Product of roots $\alpha\beta = \frac{c}{a} = k - 2 > 0 \Rightarrow k > 2$.Sum of roots $\alpha + \beta = -\frac{b}{a} = 1 > 0$,which is always true.Combining these conditions,we get $2 Therefore,the set of values for $k$ is $(2, \frac{9}{4})$.

Let $f(x) = x^{2} - x + k - 2$,where $k \in R$. Find the complete set of values of $k$ for which $y = |f(|x|)|$ is non-derivable at $5$ distinct points.

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