If `n gt 0 ` and exactly 15 integers satisfying (x+6)(x-4 ) (x-5) `(2x-n) le 0 ` then least possible value of n is

Options

A. a. 10
B. b. 12
C. c. 14
D. d. 16

Explanation

To solve the problem, we need to find the least possible value of \( n \) such that the inequality \( (x + 6)(x - 4)(x - 5)(2x - n) \leq 0 \) has exactly 15 integer solutions.

### Step-by-Step Solution:

1. **Identify the critical points**:
The critical points of the inequality are obtained by setting each factor to zero:
- \( x + 6 = 0 \) gives \( x = -6 \)
- \( x - 4 = 0 \) gives \( x = 4 \)
- \( x - 5 = 0 \) gives \( x = 5 \)
- \( 2x - n = 0 \) gives \( x = \frac{n}{2} \)

Thus, the critical points are \( -6, 4, 5, \frac{n}{2} \).

2. **Arrange the critical points on a number line**:
The order of the critical points on the number line will help us determine the intervals where the product is positive or negative. The points are:
- \( -6 \)
- \( 4 \)
- \( 5 \)
- \( \frac{n}{2} \)

3. **Determine the intervals**:
The critical points divide the number line into intervals:
- \( (-\infty, -6) \)
- \( (-6, 4) \)
- \( (4, 5) \)
- \( (5, \frac{n}{2}) \)
- \( (\frac{n}{2}, \infty) \)

4. **Test the sign of the product in each interval**:
We can test the sign of the product \( (x + 6)(x - 4)(x - 5)(2x - n) \) in each interval:
- For \( x < -6 \): All factors are negative, so the product is positive.
- For \( -6 < x < 4 \): One factor is negative, so the product is negative.
- For \( 4 < x < 5 \): Two factors are negative, so the product is positive.
- For \( 5 < x < \frac{n}{2} \): Three factors are negative, so the product is negative.
- For \( x > \frac{n}{2} \): All factors are positive, so the product is positive.

5. **Determine the intervals where the product is less than or equal to zero**:
The product is less than or equal to zero in the intervals:
- \( [-6, 4] \)
- \( [5, \frac{n}{2}] \)

6. **Count the integer solutions**:
- The integers in the interval \( [-6, 4] \) are \( -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4 \) which gives us 11 integers.
- The integers in the interval \( [5, \frac{n}{2}] \) depend on the value of \( n \).

To have exactly 15 integers satisfying the inequality, we need:
\[ 11 + \text{(number of integers in } [5, \frac{n}{2}] \text{)} = 15 \]
Thus, the number of integers in \( [5, \frac{n}{2}] \) must be 4.

7. **Solve for \( n \)**:
The number of integers in the interval \( [5, \frac{n}{2}] \) is given by:
\[ \frac{n}{2} - 5 + 1 = 4 \]
Simplifying gives:
\[ \frac{n}{2} - 4 = 4 \implies \frac{n}{2} = 8 \implies n = 16 \]

8. **Conclusion**:
The least possible value of \( n \) such that there are exactly 15 integers satisfying the inequality is \( n = 16 \).