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Determine the range of the following expressions. (x2 + x + 1)/(x2 - x + 1) |
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Answer» Let y = (x2 + x + 1)/(x2 - x + 1) ⇒ x2y – xy + y = x2 + x + 1 ⇒ x2y – xy + y – x2 – x – 1 = 0 ⇒ x2(y – 1) – x(y + 1) + (y – 1) = 0 x is real ⇒ b2 – 4ac ≥ 0 ⇒ (y + 1)2 – 4(y – 1)2 ≥ 0 ⇒ (y + 1)2 – (2y – 2)2 ≥ 0 ⇒ (y + 1 + 2y – 2) (y + 1 – 2y + 2) ≥ 0 ⇒ (3y – 1) (–y + 3) ≥ 0 ⇒ –(3y – 1) (y – 3) ≥ 0 a = coeff. of y2 = –3 < 0.,But The expression ≥ 0 ⇒ y lies between 1/3 and 3 ∴ The range of (x2 + x + 1)/(x2 - x + 1) is [1/3, 3] |
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