For any real value of x the maximum value of `8x - 3x^2` is

Options

A. `8/3`
B. 4
C. 5
D. `16/3`

Answer

To find the maximum value of the quadratic expression \( 8x - 3x^2 \), we will follow these steps:

### Step 1: Identify the coefficients
We can rewrite the expression in the standard form of a quadratic equation:
\[ -3x^2 + 8x \]
Here, we identify the coefficients:
- \( a = -3 \)
- \( b = 8 \)
- \( c = 0 \)

### Step 2: Determine if the parabola opens downwards
Since \( a < 0 \) (specifically \( a = -3 \)), the parabola opens downwards, indicating that there is a maximum value.

### Step 3: Use the formula for the maximum value of a quadratic
The maximum value of a quadratic function \( ax^2 + bx + c \) when \( a < 0 \) can be found using the formula:
\[ \text{Maximum value} = \frac{4ac - b^2}{4a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ \text{Maximum value} = \frac{4(0)(-3) - 8^2}{4(-3)} \]

### Step 4: Calculate the maximum value
Now, we compute the expression:
\[ = \frac{0 - 64}{-12} \]
\[ = \frac{-64}{-12} \]
\[ = \frac{64}{12} \]
Now, simplify \( \frac{64}{12} \):
\[ = \frac{16}{3} \]

### Conclusion
Thus, the maximum value of \( 8x - 3x^2 \) is:
\[ \frac{16}{3} \]