Question

For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`

Options

• 0

• 1

• 3

• `1/3`

Answer 1

`1/3`

Explanation:

Let `y = (1 - x + x^2)/(1 + x + x^2)`

`(dy)/(dx) = ((-1 + 2x)(1 + x + x^2) - (1x + 2x)(1 + 2x))/(1 + x + x^2)^2`

Numerator of = `(dy)/(dx) (-1 + 2x)(1 + x + x^2) - (1 + 2x)(1 - x + x^2)`

= `(-1 - x - x^2) + (2x + 2x^2 + 2x^3) - (1 - x + x^2) - (2x - 2x^2 + 2x^3)`

= `-2 + 2x^2 = 2(x^2 - 1) = 2(x - 1)(x + 1)`

∴ `(dy)/(dx) = (2(x - 1)(x + 1))/(x^2 + x + 1)^2`

`(dy)/(dx)` = 0 at `x = 1, -1`

`Ax = 1, (dy)/(dx)` Changes sign from negative to positive.

`y` is minimum at `x` = 1

Minimum value of `(1 - x + x^2)/(1 + x + x^2) = (1 - 1 + 1)/(1 + 1 + 1) = 1/3`