The greatest value of 44x2+4x+9

Let y=44x2+4x+9⇒4yx2+4xy+9y-4=0 ∴∀x∈R ⇒∆≥0 ⇒b2-4ac≥0 ⇒16y2-4(4y)(9y-4)≥0 ⇒y2-9y2+4y=0 ⇒8y2-4y≤0 ⇒4y2y-1≤0 ⇒y2y-1≤0 ⇒y∈0 , 12

The greatest value of 44x2+4x+9

Let y=44x2+4x+9⇒4yx2+4xy+9y-4=0 ∴∀x∈R ⇒∆≥0 ⇒b2-4ac≥0 ⇒16y2-4(4y)(9y-4)≥0 ⇒y2-9y2+4y=0 ⇒8y2-4y≤0 ⇒4y2y-1≤0 ⇒y2y-1≤0 ⇒y∈0 , 12

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The greatest value of $\frac{4}{4 x^{2} + 4 x + 9}$

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$\frac{9}{4}$

$\begin{array}{l} 1 \\ \bar{2} \end{array}$

$\begin{array}{l} 4 \\ \bar{9} \end{array}$

$4$

answer is D.

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Detailed Solution

$L e t y = \frac{4}{4 x^{2} + 4 x + 9}$

$\Rightarrow 4 y x^{2} + 4 x y + 9 y - 4 = 0 ∴ \forall x \in R \Rightarrow ∆ \geq 0 \Rightarrow b^{2} - 4 a c \geq 0 \Rightarrow 16 y^{2} - 4 (4 y) (9 y - 4) \geq 0 \Rightarrow y^{2} - 9 y^{2} + 4 y = 0 \Rightarrow 8 y^{2} - 4 y \leq 0 \Rightarrow 4 y (2 y - 1) \leq 0 \Rightarrow y (2 y - 1) \leq 0 \Rightarrow y \in [0, \frac{1}{2}]$