The greatest value of `(4)/(4x^(2)+4x+9)` is

Options

A. `(4)/(9)`
B. 4
C. `(9)/(4)`
D. `(1)/(2)`

Answer

To find the greatest value of the expression \( \frac{4}{4x^2 + 4x + 9} \), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression:
\[ \frac{4}{4x^2 + 4x + 9} \]
We can factor out a 4 from the denominator:
\[ = \frac{4}{4(x^2 + x + \frac{9}{4})} = \frac{1}{x^2 + x + \frac{9}{4}} \]

### Step 2: Complete the Square
Next, we will complete the square for the quadratic expression in the denominator:
\[ x^2 + x + \frac{9}{4} \]
To complete the square, we take the coefficient of \(x\), which is 1, divide it by 2 to get \(\frac{1}{2}\), and square it to get \(\frac{1}{4}\). We can rewrite the expression as:
\[ = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} + \frac{9}{4} \]
This simplifies to:
\[ = \left(x + \frac{1}{2}\right)^2 + 2 \]

### Step 3: Rewrite the Expression
Now, we can rewrite our original expression:
\[ \frac{1}{\left(x + \frac{1}{2}\right)^2 + 2} \]

### Step 4: Find the Minimum Value of the Denominator
The expression \(\left(x + \frac{1}{2}\right)^2\) is always non-negative and reaches its minimum value of 0 when \(x = -\frac{1}{2}\). Thus, the minimum value of the denominator is:
\[ 0 + 2 = 2 \]

### Step 5: Find the Maximum Value of the Original Expression
Since we want to maximize \(\frac{1}{\text{denominator}}\), we find:
\[ \text{Maximum value} = \frac{1}{2} \]

### Conclusion
Thus, the greatest value of \(\frac{4}{4x^2 + 4x + 9}\) is:
\[ \frac{1}{2} \]