If `f(x)=x^(3) -4x +p`, and f (0) and f (1) are of opposite signs, then which of the following is necessarily true?

Options

A. `-1 lt p lt 2`
B. `0 lt p lt 3`
C. `-2 lt p lt 1`
D. `-3 lt p lt 0`

Answer

To solve the problem, we need to analyze the function \( f(x) = x^3 - 4x + p \) and find the values of \( p \) such that \( f(0) \) and \( f(1) \) have opposite signs.

### Step-by-Step Solution:

1. **Calculate \( f(0) \)**:
\[ f(0) = 0^3 - 4 \cdot 0 + p = p \]

2. **Calculate \( f(1) \)**:
\[ f(1) = 1^3 - 4 \cdot 1 + p = 1 - 4 + p = p - 3 \]

3. **Set up the condition for opposite signs**:
Since \( f(0) \) and \( f(1) \) are of opposite signs, we have two cases:
- Case 1: \( f(0) > 0 \) and \( f(1) < 0 \)
- Case 2: \( f(0) < 0 \) and \( f(1) > 0 \)

This gives us the following inequalities:
- For Case 1: \( p > 0 \) and \( p - 3 < 0 \)
- For Case 2: \( p < 0 \) and \( p - 3 > 0 \)

4. **Solve the inequalities**:
- From Case 1:
- \( p > 0 \)
- \( p < 3 \)
- Thus, \( 0 < p < 3 \)

- From Case 2:
- \( p < 0 \)
- \( p > 3 \)
- This case is impossible since \( p \) cannot be both less than 0 and greater than 3 at the same time.

5. **Conclusion**:
The only valid case is Case 1, which gives us the range:
\[ 0 < p < 3 \]
Therefore, the values of \( p \) must lie in the interval \( (0, 3) \).

### Final Answer:
The necessary condition is that \( p \) lies in the range \( (0, 3) \).

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