If `f(x)` = `x^(3)` +` b``x^(2)` + `c``x`+`d` and `0 lt b^2 lt c,` then in `(-oo,oo), f (x)`

Options

A. is increasing
B. has local maxima
C. is decreasing
D. is bounded

Answer

To determine the behavior of the function \( f(x) = x^3 + bx^2 + cx + d \) under the condition \( 0 < b^2 < c \), we will analyze the first derivative of the function.

### Step-by-Step Solution:

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

2. **Identify the coefficients**:
The first derivative \( f'(x) \) is a quadratic polynomial of the form \( ax^2 + bx + c \):
- Here, \( a = 3 \), \( b = 2b \), and \( c = c \).

3. **Determine the discriminant**:
The discriminant \( D \) of a quadratic \( ax^2 + bx + c \) is given by:
\[ D = (2b)^2 - 4 \cdot 3 \cdot c = 4b^2 - 12c \]

4. **Analyze the condition**:
Given \( 0 < b^2 < c \), we can infer:
- Since \( b^2 < c \), it follows that \( 12c > 12b^2 \).
- Therefore, \( 4b^2 - 12c < 0 \) implies \( D < 0 \).

5. **Conclusion about the first derivative**:
Since \( D < 0 \) and \( a = 3 > 0 \):
- The quadratic \( f'(x) = 3x^2 + 2bx + c \) has no real roots and opens upwards.
- This means \( f'(x) > 0 \) for all \( x \in (-\infty, \infty) \).

6. **Final conclusion about the function**:
Since \( f'(x) > 0 \) for all \( x \), the function \( f(x) \) is strictly increasing on the entire real line \( (-\infty, \infty) \).

### Summary:
Thus, we conclude that the function \( f(x) = x^3 + bx^2 + cx + d \) is an increasing function on the interval \( (-\infty, \infty) \).