Quadratic Equations and Inequalities: Minimum expression MCQ

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Minimum expressionHardQuestion 220044

Question

For real x, the minimum value of (x²+x+1)/(x²−x+1) is:

A
1/3
Correct
B
1
C
3
D
0

Correct answer

1/3

Explanation

Let y = (x²+x+1)/(x²−x+1). Then yx²−yx+y = x²+x+1 → (y−1)x²−(y+1)x+(y−1) = 0. For real x: D ≥ 0 → (y+1)² − 4(y−1)² ≥ 0 → 3y²−10y+3 ≤ 0 → (3y−1)(y−3) ≤ 0 → 1/3 ≤ y ≤ 3. Minimum = 1/3.

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