The set of value of `a` for which `(a-1)x^2(a+1)x+a-1geq0` is true for all `xgeq2` is (a) `(-oo,1)` b. `(1,7/3)` c. `(7/3,oo)` d. none of these

Options

A. `(-infty, 1)`
B. `(1, (7)/(3))`
C. `((7)/(3), infty)`
D. none of these

Answer

3

Explanation

Given,
`(a - 1) x^(2) - (a + 1) x + a - 1 ge 0 `
or ` a(x^(2) - x + 1) - (x^(2) + x + 1) ge 0 `
` rArr ge (x^(2) + x + 1)/(x^(2) - x + 1) = (2x)/(x^(2) - x + 1)`
` = 1 + (2)/(x + (1)/(x) - 1)`
Let y = x + 1/x , Now , y is increasing in ` [ 2, infty)` . Hence,
` = 1 + (2)/(x + (1)/(x) - 1)in (1 , (7)/(3))`
For all ` x ge 2,` Eq (1). should be true . Hence , ` a ge 7//3)`