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

Options

A. f(x) is a strictly increasing function
B. f(x) has a local maxima
C. f(x) is a strictly decreasing function
D. f(x) is bounded

Answer

To solve the problem step by step, we start with the given function:

### Step 1: Write down the function
The function is given as:
\[ f(x) = x^3 + bx^2 + cx + d \]

### Step 2: Differentiate the function
We need to find the first derivative of the function to analyze its behavior:
\[ f'(x) = \frac{d}{dx}(x^3 + bx^2 + cx + d) \]
Using the power rule:
\[ f'(x) = 3x^2 + 2bx + c \]

### Step 3: Analyze the first derivative
To determine whether the function is increasing or decreasing, we need to analyze the first derivative \( f'(x) \). We will check if \( f'(x) \) can be zero or if it is always positive or negative.

### Step 4: Find the discriminant of the quadratic
The expression \( f'(x) = 3x^2 + 2bx + c \) is a quadratic equation in \( x \). The discriminant \( D \) of a quadratic \( ax^2 + bx + c \) is given by:
\[ D = B^2 - 4AC \]
For our derivative:
- \( A = 3 \)
- \( B = 2b \)
- \( C = c \)

Thus, the discriminant is:
\[ D = (2b)^2 - 4(3)(c) = 4b^2 - 12c \]

### Step 5: Analyze the discriminant
We know from the problem statement that \( 0 < b^2 < c \). This implies:
\[ 4b^2 - 12c < 0 \]
This means the discriminant \( D < 0 \), indicating that the quadratic \( 3x^2 + 2bx + c \) has no real roots and does not cross the x-axis.

### Step 6: Determine the sign of the derivative
Since the leading coefficient (3) of the quadratic \( 3x^2 + 2bx + c \) is positive and the discriminant is negative, this means that the quadratic is always positive:
\[ f'(x) > 0 \text{ for all } x \]

### Step 7: Conclusion about the function
Since \( f'(x) > 0 \) for all \( x \), the function \( f(x) \) is strictly increasing on the interval \( (-\infty, \infty) \). Therefore, it does not have any local maxima or minima.

### Final Answer
The function \( f(x) \) is strictly increasing on \( (-\infty, \infty) \).

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