If x 2 + m x + 1 x 2 + x + 1 3 for all real x , then

Question

If x 2 + m x + 1 x 2 + x + 1 3 for all real x , then, m - 1, - 1 m 6, - 1 m 5, m > 6

MARKS App · Accepted Answer

We have x 2 + x + 1 = x + 1 2 2 + 3 4 > 0 So, - 3 x 2 + m x + 1 x 2 + x + 1 3 ⇒ - 3 x 2 + x + 1 x 2 + m x + 1 3 x 2 + x + 1 ⇒ 4 x 2 + m + 3 x + 4 > 0 and 2 x 2 + 3 - m x + 2 > 0 for real x ⇒ m + 3 2 - 4 × 4 × 4 0 and 3 - m 2 - 4 × 2 × 2 0 ⇒ m + 3 + 8 m + 3 - 8 0 and m - 3 + 4 m - 3 - 4 0 ⇒ m + 11 m - 5 0 and m + 1 m - 7 0 ⇒ - 11 m 5 and - 1 m 7 Hence, - 1 m 5

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