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Question: If x ∈ R, then \(\frac{x^{2} - x + 1}{x^{2} + x + 1}\)takes values in the interval –...

If x ∈ R, then x2−x+1x2+x+1\frac{x^{2} - x + 1}{x^{2} + x + 1}takes values in the interval –

A

(13,3)\left( \frac{1}{3},3 \right)

B

[13,3]\left\lbrack \frac{1}{3},3 \right\rbrack

C

(0, 3)

D

None of these

Answer

[13,3]\left\lbrack \frac{1}{3},3 \right\rbrack

Explanation

Solution

Let y = x2−x+1x2+x+1\frac{x^{2} - x + 1}{x^{2} + x + 1} So, D ≥ 0

Q (1 + y)2 – 4 (1 – y)2 ≥ 0

⇒ – (y – 3) (y−13)\left( y - \frac{1}{3} \right)≤ 0; y ∈ [13,3]\left\lbrack \frac{1}{3},3 \right\rbrack