Suppose `f(x)=(1)/(2)x^(2)-8` for `-4lexle4`. Then the maximum value of the graph of `|f(x)|` is

Options

A. `-8`
B. 0
C. 2
D. 8

Explanation

To find the maximum value of the graph of \( |f(x)| \) where \( f(x) = \frac{1}{2}x^2 - 8 \) for \( -4 \leq x \leq 4 \), we will follow these steps:

### Step 1: Determine the range of \( f(x) \)

1. **Find the values of \( f(x) \) at the endpoints of the interval**:
- Calculate \( f(-4) \):
\[ f(-4) = \frac{1}{2}(-4)^2 - 8 = \frac{1}{2}(16) - 8 = 8 - 8 = 0 \]
- Calculate \( f(4) \):
\[ f(4) = \frac{1}{2}(4)^2 - 8 = \frac{1}{2}(16) - 8 = 8 - 8 = 0 \]

2. **Find the vertex of the parabola**:
- The function \( f(x) = \frac{1}{2}x^2 - 8 \) is a parabola that opens upwards. The vertex form can be found using the formula \( x = -\frac{b}{2a} \):
\[ x = -\frac{0}{2 \cdot \frac{1}{2}} = 0 \]
- Calculate \( f(0) \):
\[ f(0) = \frac{1}{2}(0)^2 - 8 = -8 \]

3. **Determine the range of \( f(x) \)**:
- From the calculations, we have:
- \( f(-4) = 0 \)
- \( f(4) = 0 \)
- \( f(0) = -8 \)
- Therefore, the range of \( f(x) \) for \( -4 \leq x \leq 4 \) is:
\[ -8 \leq f(x) \leq 0 \]

### Step 2: Determine the range of \( |f(x)| \)

1. **Calculate \( |f(x)| \)**:
- Since \( f(x) \) ranges from \(-8\) to \(0\), the absolute value \( |f(x)| \) will range from:
\[ |f(x)| = \begin{cases} -f(x) & \text{if } f(x) < 0 \\ f(x) & \text{if } f(x) \geq 0 \end{cases} \]
- Thus, the range of \( |f(x)| \) will be from \(0\) to \(8\).

### Step 3: Find the maximum value of \( |f(x)| \)

1. **Identify the maximum value**:
- The maximum value of \( |f(x)| \) is \(8\), which occurs when \( f(x) \) is at its minimum value of \(-8\).

### Conclusion

The maximum value of the graph of \( |f(x)| \) is:
\[ \boxed{8} \]