The minimum value of the expression `|x-p|+|x-15|+|x-p-15|` for `' x '` in the range `plt=xlt=15` where `x

Options

A. 10
B. 15
C. 30
D. 0

Answer

To find the minimum value of the expression \( |x - p| + |x - 15| + |x - p - 15| \) for \( x \) in the range \( p < x < 15 \), we can follow these steps:

### Step 1: Understand the expression
The expression consists of three absolute value terms. The absolute value function has different behaviors depending on the value of \( x \) relative to \( p \) and \( 15 \).

### Step 2: Analyze the intervals
Since \( p < x < 15 \), we need to consider the behavior of the absolute value terms in this interval. The critical points where the expression may change are at \( x = p \) and \( x = 15 \).

### Step 3: Break down the expression based on intervals
1. **For \( x < p \)**: This case does not apply since \( x \) cannot be less than \( p \).
2. **For \( p \leq x < 15 \)**: Here, we need to evaluate the expression:
- \( |x - p| = x - p \) (since \( x \geq p \))
- \( |x - 15| = 15 - x \) (since \( x < 15 \))
- \( |x - p - 15| = x - p - 15 \) (since \( x < p + 15 \))

Therefore, the expression simplifies to:
\[ (x - p) + (15 - x) + (x - p - 15) = -p + x - p - 15 + 15 = -2p + x \]

### Step 4: Find the minimum value
Now, we have the expression \( -2p + x \). To minimize this expression within the range \( p < x < 15 \), we should consider the maximum value of \( x \) since \( -2p \) is constant for a given \( p \).

Thus, the minimum value occurs when \( x \) approaches \( 15 \):
\[ \text{Minimum value} = -2p + 15 \]

### Step 5: Determine the minimum value based on \( p \)
Since \( p < 15 \), we can substitute different values of \( p \) to find the minimum value of the expression. However, we need to ensure that \( p \) is less than \( 15 \) and greater than \( 0 \).

### Conclusion
The minimum value occurs when \( p \) is as small as possible (approaching \( 0 \)), leading to:
\[ \text{Minimum value} = 15 - 2p \to 15 \text{ as } p \to 0 \]

Thus, the minimum value of the expression is **15**.