Let `f(x)=a x^3+b x^2+c x+d , a!=0` If `x_1 and x_2` are the real and distinct roots of `f prime(x)=0` then `f(x)=0` will have three real and distinct roots if

Options

A. `x_(1). x_(2) lt 0`
B. `f (x_(1)). f (x_(2)) gt 0`
C. `f (x_(1)). f (x_(2)) lt 0`
D. `x_(2), x_(2) gt 0`

Answer

To solve the problem, we need to analyze the function \( f(x) = ax^3 + bx^2 + cx + d \) and its derivative \( f'(x) \). We are given that \( f'(x) = 0 \) has two distinct real roots \( x_1 \) and \( x_2 \). We want to find the condition under which \( f(x) = 0 \) has three distinct real roots.

### Step-by-Step Solution:

1. **Find the derivative of \( f(x) \)**:
\[ f'(x) = \frac{d}{dx}(ax^3 + bx^2 + cx + d) = 3ax^2 + 2bx + c \]

2. **Set the derivative equal to zero**:
\[ f'(x) = 0 \implies 3ax^2 + 2bx + c = 0 \]
Since \( x_1 \) and \( x_2 \) are the distinct real roots of this quadratic equation, the discriminant must be positive:
\[ D = (2b)^2 - 4(3a)(c) > 0 \implies 4b^2 - 12ac > 0 \]

3. **Analyze the behavior of \( f(x) \)**:
Since \( f'(x) \) has two distinct real roots, the graph of \( f(x) \) will have a local maximum and a local minimum. For \( f(x) \) to have three distinct real roots, the function must cross the x-axis three times.

4. **Evaluate \( f(x) \) at the critical points**:
Let \( f(x_1) \) and \( f(x_2) \) be the values of the function at the critical points \( x_1 \) and \( x_2 \). For \( f(x) \) to have three distinct roots, the following condition must hold:
\[ f(x_1) \cdot f(x_2) < 0 \]
This means that one of \( f(x_1) \) or \( f(x_2) \) must be positive and the other must be negative.

5. **Conclusion**:
Therefore, the condition for \( f(x) = 0 \) to have three distinct real roots is:
\[ f(x_1) \cdot f(x_2) < 0 \]
This condition ensures that the cubic function crosses the x-axis three times.