The number of integral values of `a` for which `f(x) = x^3 + (a + 2)x^2 + 3ax + 5` is monotonic in `AA x in R`

Options

A. 2
B. 4
C. 6
D. 7

Answer

To determine the number of integral values of \( a \) for which the function \( f(x) = x^3 + (a + 2)x^2 + 3ax + 5 \) is monotonic for \( x \in \mathbb{R} \), we need to analyze the derivative of the function.

### Step-by-Step Solution:

1. **Find the Derivative**:
The first step is to compute the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^3 + (a + 2)x^2 + 3ax + 5) \]
Applying the power rule:
\[ f'(x) = 3x^2 + 2(a + 2)x + 3a \]

2. **Determine Monotonicity**:
For \( f(x) \) to be monotonic (either always increasing or always decreasing), \( f'(x) \) must not change sign. This means that the quadratic \( f'(x) \) should either be always non-negative or always non-positive.

3. **Analyze the Quadratic**:
The quadratic \( f'(x) = 3x^2 + 2(a + 2)x + 3a \) will be non-negative for all \( x \) if its discriminant is less than or equal to zero. The discriminant \( D \) of a quadratic \( Ax^2 + Bx + C \) is given by:
\[ D = B^2 - 4AC \]
Here, \( A = 3 \), \( B = 2(a + 2) \), and \( C = 3a \). Thus,
\[ D = [2(a + 2)]^2 - 4 \cdot 3 \cdot 3a \]
Simplifying this:
\[ D = 4(a + 2)^2 - 36a \]
\[ = 4(a^2 + 4a + 4) - 36a \]
\[ = 4a^2 + 16a + 16 - 36a \]
\[ = 4a^2 - 20a + 16 \]

4. **Set the Discriminant Less Than or Equal to Zero**:
We need to solve the inequality:
\[ 4a^2 - 20a + 16 \leq 0 \]
Dividing the entire inequality by 4:
\[ a^2 - 5a + 4 \leq 0 \]

5. **Factor the Quadratic**:
Factoring the quadratic:
\[ (a - 1)(a - 4) \leq 0 \]

6. **Find the Intervals**:
The critical points are \( a = 1 \) and \( a = 4 \). We analyze the sign of the expression in the intervals:
- For \( a < 1 \): The expression is positive.
- For \( 1 \leq a \leq 4 \): The expression is non-positive.
- For \( a > 4 \): The expression is positive.

Thus, the solution to the inequality is:
\[ 1 \leq a \leq 4 \]

7. **Determine Integral Values**:
The integral values of \( a \) in the interval \( [1, 4] \) are \( 1, 2, 3, 4 \). Therefore, there are 4 integral values.

### Final Answer:
The number of integral values of \( a \) for which \( f(x) \) is monotonic is **4**.

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