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. 30
B. 15
C. 10
D. 0

Answer

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

### Step 1: Analyze the expression
We start with the expression:
\[ E(x) = |x - p| + |x - 15| + |x - p - 15| \]

### Step 2: Substitute \( x = p \)
Let's evaluate the expression at the point \( x = p \):
\[ E(p) = |p - p| + |p - 15| + |p - p - 15| \]
This simplifies to:
\[ E(p) = 0 + |p - 15| + |-15| = |p - 15| + 15 \]
Since \( p < 15 \), we have \( |p - 15| = 15 - p \). Therefore:
\[ E(p) = (15 - p) + 15 = 30 - p \]

### Step 3: Determine the range of \( p \)
Given that \( 0 < p < 15 \), we can analyze \( E(p) \):
- As \( p \) increases from 0 to 15, \( 30 - p \) decreases from 30 to 15.

### Step 4: Evaluate at the endpoints
Next, we need to evaluate \( E(x) \) at the endpoint \( x = 15 \):
\[ E(15) = |15 - p| + |15 - 15| + |15 - p - 15| \]
This simplifies to:
\[ E(15) = |15 - p| + 0 + |0 - p| = |15 - p| + | - p| = (15 - p) + p = 15 \]

### Step 5: Compare values
Now we compare the values:
- At \( x = p \): \( E(p) = 30 - p \)
- At \( x = 15 \): \( E(15) = 15 \)

### Step 6: Conclusion
Since \( 30 - p \) decreases as \( p \) increases and is always greater than or equal to 15 for \( p < 15 \), the minimum value of the expression occurs at \( x = 15 \):
\[ \text{Minimum value} = 15 \]

### Final Answer
The minimum value of the expression is \( \boxed{15} \).