AP EAMCET · Maths · Quadratic Equation
The maximum value of the expression \(\frac{x^2+x+1}{2 x^2-x+1}\), for \(x \in \mathbb{R}\), is
- A \(\frac{7+2 \sqrt{7}}{7}\)
- B \(\frac{7-2 \sqrt{7}}{7}\)
- C \(\frac{7}{3}\)
- D \(\frac{14+2 \sqrt{7}}{7}\)
Answer & Solution
Correct Answer
(A) \(\frac{7+2 \sqrt{7}}{7}\)
Step-by-step Solution
Detailed explanation
Let \(y = \frac{x^2+x+1}{2 x^2-x+1}\) \((2y-1)x^2 - (y+1)x + (y-1) = 0\) For real \(x\), the discriminant \(D \ge 0\): \((- (y+1))^2 - 4(2y-1)(y-1) \ge 0\) \(y^2+2y+1 - 4(2y^2-3y+1) \ge 0\) \(y^2+2y+1 - 8y^2+12y-4 \ge 0\) \(-7y^2 + 14y - 3 \ge 0\) \(7y^2 - 14y + 3 \le 0\) Roots…
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