The number of possible integral values of the parameter kk for which the inequality k^2x^2 < (8k -3)(x+6)k2x2<(8k−3)(x+6) holds true for all values of xx satisfying x^2 < x+2x2<x+2 is?

A) 0
B) 1
C) 2
D) 3

1 Answer
Cesareo R.
Nov 27, 2017

00

Explanation:

x^2 < x + 2x2<x+2 is true for x in (-1,2)x∈(−1,2)

now solving for kk

k^2 x^2 - (8 k - 3) (x + 6) < 0k2x2−(8k−3)(x+6)<0 we have

k in ((24 + 4 x - sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2, (24 + 4 x + sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2)k∈(24+4x−√242+192x−2x2−3x3x2,24+4x+√242+192x−2x2−3x3x2)

but

(24 + 4 x + sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^224+4x+√242+192x−2x2−3x3x2 is unbounded as xx approaches 00 so the answer is 00 integer values for kk obeying the two conditions.

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