If absolute maximum value of
`f(x)=(1)/(|x-4|+1)+(1)/(|x+8|+1)is(p)/(q),` (p,q are coprime) the (p-q) is……… .

Answer

To solve the problem of finding the absolute maximum value of the function

\[ f(x) = \frac{1}{|x-4|+1} + \frac{1}{|x+8|+1} \]

and determining \(p - q\) where the maximum value is expressed as \(\frac{p}{q}\) (with \(p\) and \(q\) being coprime), we will follow these steps:

### Step 1: Analyze the Function
The function consists of two parts, both of which involve absolute values. We need to consider the critical points where the expressions inside the absolute values change, which are \(x = 4\) and \(x = -8\).

### Step 2: Evaluate the Function at Critical Points
We will evaluate \(f(x)\) at the critical points and also check the behavior of the function as \(x\) approaches these points.

1. **At \(x = 4\)**:
\[ f(4) = \frac{1}{|4-4|+1} + \frac{1}{|4+8|+1} = \frac{1}{0+1} + \frac{1}{12+1} = 1 + \frac{1}{13} = 1 + \frac{1}{13} = \frac{14}{13} \]

2. **At \(x = -8\)**:
\[ f(-8) = \frac{1}{|-8-4|+1} + \frac{1}{|-8+8|+1} = \frac{1}{12+1} + \frac{1}{0+1} = \frac{1}{13} + 1 = \frac{1}{13} + \frac{13}{13} = \frac{14}{13} \]

### Step 3: Check Behavior as \(x\) Approaches Infinity
As \(x\) approaches positive or negative infinity, both terms in \(f(x)\) approach zero:
\[ \lim_{x \to \pm \infty} f(x) = 0 \]

### Step 4: Determine Maximum Value
From the evaluations at the critical points, we find:
- \(f(4) = \frac{14}{13}\)
- \(f(-8) = \frac{14}{13}\)

Thus, the absolute maximum value of \(f(x)\) is \(\frac{14}{13}\).

### Step 5: Identify \(p\) and \(q\)
Here, \(p = 14\) and \(q = 13\). Since \(14\) and \(13\) are coprime, we can proceed to calculate \(p - q\).

### Step 6: Calculate \(p - q\)
\[ p - q = 14 - 13 = 1 \]

### Final Answer
The value of \(p - q\) is \(1\).

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