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Question Number 33997 by rahul 19 last updated on 29/Apr/18

$$\mathit{I}\mathrm{f}\mathrm{the}\mathrm{range}\mathrm{of}\mathrm{the}\mathrm{function}$$$$\mathrm{f}\left(x\right)=\frac{x-1}{\mathrm{p}-x^2+1}\mathrm{does}\mathrm{not}\mathrm{contain}\mathrm{any}$$$$\mathrm{values}\mathrm{belonging}\mathrm{to}\mathrm{the}\mathrm{interval}$$$$[-1,\frac{-1}{3}]thentrueset\mathrm{of}\mathrm{values}\mathrm{of}\mathrm{p}\mathrm{is}?$$

Answered by ajfour last updated on 29/Apr/18

$$for{x}^2>p+1$$$$\frac{x-1}{p+1-x^2}<-1$$$$\Rightarrow x^2-p-1<x-1$$$$x^2-x-p<0$$$$\Rightarrow novalueofp.$$$$if\frac{x-1}{p+1-x^2}>-\frac{1}{3}$$$$x^2-p-1>3x-3$$$$x^2-3x-p+2>0\left(forallx\right)$$$$\Rightarrow 9+4p-8<0$$$$\Rightarrow p<-\frac{1}{4}$$$$if{x}^2<p+1$$$$then{x}^2-x+p>0$$$$\Rightarrow 1-4p<0$$$$\Rightarrow p>\frac{1}{4}$$$$similarly$$$$x^2-3x-p+2<0$$$$novalue.$$$$\mathit{h}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathit{p}\mathit{o}\mathit{s}\mathit{s}\mathit{i}\mathit{b}\mathit{l}\mathit{e}\mathit{o}\mathit{n}\mathit{l}\mathit{y}\mathit{f}\mathit{o}\mathit{r}$$$$\mathit{p}\in (-\mathrm{\infty },-\frac{1}{4}).$$

Commented by rahul 19 last updated on 29/Apr/18

$$\mathcal{T}\mathrm{hank}\mathrm{you}\mathrm{sir}!$$

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