If-the-range-of-the-function-f-x-x-1-p-x-2-1-does-not-contain-any-values-belonging-to-the-interval-1-1-3-then-true-set-of-values-of-p-is-

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

I f the range of the function

f (x) = \frac{x - 1}{p - x^{2} + 1} does not contain any

values belonging to the interval

[- 1, \frac{- 1}{3}] t h e n t r u e s e t of values of p is ?

Answered by ajfour last updated on 29/Apr/18

f o r 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 n o v a l u e o f p .

i f \frac{x - 1}{p + 1 - x^{2}} > - \frac{1}{3}

x^{2} - p - 1 > 3 x - 3

x^{2} - 3 x - p + 2 > 0 (f o r a l l x)

\Rightarrow 9 + 4 p - 8 < 0

\Rightarrow p < - \frac{1}{4}

i f x^{2} < p + 1

t h e n x^{2} - x + p > 0

\Rightarrow 1 - 4 p < 0

\Rightarrow p > \frac{1}{4}

s i m i l a r l y

x^{2} - 3 x - p + 2 < 0

n o v a l u e .

h e n c e p o s s i b l e o n l y f o r

p \in (- \infty, - \frac{1}{4}) .

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

T hank you sir!