For real x, the greatest value of (x2+2x+4/2x2+4x+9) is

Question

For real x, the greatest value of (x2+2x+4/2x2+4x+9) is (A) 1 (B) -1 (C) (1/2) (D) (1/4)

Tardigrade Admin · Accepted Answer

Let y=(x2+2 x+4/2 x2+4 x+9) ⇒ 2 x2 y+4 x y+9 y=x2+2 x+4 ⇒ (2 y-1) x2+(4 y-2) x+9 y-4=0 ⇒ x=(-(4 y-2) ± √-4 y-2)2-4(2 y-1)(9 y-4)/2(2 y-1)) Since, x is real number. ∴ (4 y-2)2-4(2 y-1)(9 y-4) ≥ 0 ⇒ 4(2 y-1)2-4(2 y-1)(9 y-4) ≥ 0 ⇒ 4(2 y-1)(2 y-1-9 y+4) ≥ 0 ⇒ 4(2 y-1)(3-7 y) ≥ 0 ⇒ (2 y-1)(7 y-3) ≤ 0

Q. For real $x$ , the greatest value of $\frac{x ^{2} + 2 x + 4}{2 x ^{2} + 4 x + 9}$ is

$1$

$- 1$

$\frac{1}{2}$

$\frac{1}{4}$

Q. For real x, the greatest value of 2x2+4x+9x2+2x+4​ is

1

−1

21​

41​

Q. For real $x$ , the greatest value of $\frac{x ^{2} + 2 x + 4}{2 x ^{2} + 4 x + 9}$ is

$1$

$- 1$

$\frac{1}{2}$

$\frac{1}{4}$