If p is an integer such that `-5ltplt5 and q=3p-p^(3)`, what is the least possible value of q?

Options

A. `-76`
B. `-52`
C. `-4`
D. `0`

Answer

To find the least possible value of \( q \) given the equation \( q = 3p - p^3 \) and the constraint \( -5 < p < 5 \) where \( p \) is an integer, we can follow these steps:

### Step 1: Identify the possible integer values for \( p \)
Since \( p \) is an integer and must satisfy the inequality \( -5 < p < 5 \), the possible integer values for \( p \) are:
\[ p = -4, -3, -2, -1, 0, 1, 2, 3, 4 \]

### Step 2: Calculate \( q \) for each possible value of \( p \)
We will substitute each integer value of \( p \) into the equation \( q = 3p - p^3 \) and calculate \( q \).

- For \( p = -4 \):
\[ q = 3(-4) - (-4)^3 = -12 - (-64) = -12 + 64 = 52 \]

- For \( p = -3 \):
\[ q = 3(-3) - (-3)^3 = -9 - (-27) = -9 + 27 = 18 \]

- For \( p = -2 \):
\[ q = 3(-2) - (-2)^3 = -6 - (-8) = -6 + 8 = 2 \]

- For \( p = -1 \):
\[ q = 3(-1) - (-1)^3 = -3 - (-1) = -3 + 1 = -2 \]

- For \( p = 0 \):
\[ q = 3(0) - (0)^3 = 0 - 0 = 0 \]

- For \( p = 1 \):
\[ q = 3(1) - (1)^3 = 3 - 1 = 2 \]

- For \( p = 2 \):
\[ q = 3(2) - (2)^3 = 6 - 8 = -2 \]

- For \( p = 3 \):
\[ q = 3(3) - (3)^3 = 9 - 27 = -18 \]

- For \( p = 4 \):
\[ q = 3(4) - (4)^3 = 12 - 64 = -52 \]

### Step 3: Determine the least value of \( q \)
Now, we will compile the calculated values of \( q \):
- For \( p = -4 \), \( q = 52 \)
- For \( p = -3 \), \( q = 18 \)
- For \( p = -2 \), \( q = 2 \)
- For \( p = -1 \), \( q = -2 \)
- For \( p = 0 \), \( q = 0 \)
- For \( p = 1 \), \( q = 2 \)
- For \( p = 2 \), \( q = -2 \)
- For \( p = 3 \), \( q = -18 \)
- For \( p = 4 \), \( q = -52 \)

The least value of \( q \) is \( -52 \) when \( p = 4 \).

### Final Answer
The least possible value of \( q \) is:
\[ \boxed{-52} \]