The function `f(x) = abs( px - q) + r absx, x in (-oo,oo)` where `p gt 0, q gt 0, r gt 0`, assumes its minimum value only on one point if

Options

A. `pne q`
B. `r ne q`
C. `r ne p`
D. `p = q=r`

Answer

To find the condition under which the function \( f(x) = |px - q| + r|x| \) assumes its minimum value at only one point, we can analyze the function based on the properties of absolute values and the behavior of the function in different intervals.

### Step 1: Identify the critical points

The function \( f(x) \) involves absolute values, which means we need to consider the points where the expressions inside the absolute values change sign. These points are:

1. \( px - q = 0 \) → \( x = \frac{q}{p} \)
2. \( x = 0 \)

Thus, the critical points are \( x = 0 \) and \( x = \frac{q}{p} \).

### Step 2: Analyze the function in different intervals

We will analyze the function in three intervals based on the critical points:

1. **Interval 1:** \( x < 0 \)
- Here, \( |px - q| = q - px \) and \( |x| = -x \)
- Therefore, \( f(x) = q - px - rx = q - (p + r)x \)

2. **Interval 2:** \( 0 \leq x < \frac{q}{p} \)
- Here, \( |px - q| = q - px \) and \( |x| = x \)
- Thus, \( f(x) = q - px + rx = q + (r - p)x \)

3. **Interval 3:** \( x \geq \frac{q}{p} \)
- Here, \( |px - q| = px - q \) and \( |x| = x \)
- Therefore, \( f(x) = px - q + rx = (p + r)x - q \)

### Step 3: Determine the slopes in each interval

- **Interval 1:** The slope is \( -(p + r) \) (negative).
- **Interval 2:** The slope is \( r - p \).
- **Interval 3:** The slope is \( p + r \) (positive).

### Step 4: Find conditions for a single minimum point

For \( f(x) \) to have a minimum at only one point, the slopes must behave such that:

1. In **Interval 1**, the function is decreasing.
2. In **Interval 2**, the function must either be constant or increasing. If \( r - p < 0 \) (i.e., \( r < p \)), the function will continue to decrease, leading to a minimum at \( x = 0 \).
3. In **Interval 3**, the function must be increasing.

Thus, for the function to have a minimum at only one point, we need:

- \( r - p = 0 \) (i.e., \( r = p \)) to ensure that the function is constant at \( x = 0 \) and then starts increasing in the third interval.

### Conclusion

The function \( f(x) = |px - q| + r|x| \) assumes its minimum value only at one point if \( r = p \).