The possible value of the ordered triplet (a, b, c) such that the function `f(x)=x^(3)+ax^(2)+bx+c` is a monotonic function is

Options

A. `(2, 1, 3)`
B. `(1, -1, 3)`
C. `(2, 2, 4)`
D. `(2, -2, 1)`

Answer

To determine the possible values of the ordered triplet (a, b, c) such that the function \( f(x) = x^3 + ax^2 + bx + c \) is a monotonic function, we need to follow these steps:

### Step 1: Understand the condition for monotonicity
A function is monotonic if its derivative does not change sign. This means that the derivative must either be always positive or always negative.

### Step 2: Differentiate the function
We differentiate \( f(x) \):
\[ f'(x) = 3x^2 + 2ax + b \]

### Step 3: Set the condition for monotonicity
For \( f(x) \) to be monotonic, \( f'(x) \) must not equal zero for any real number \( x \). This implies that the quadratic equation \( 3x^2 + 2ax + b = 0 \) should have no real roots.

### Step 4: Use the discriminant condition
The condition for a quadratic equation \( Ax^2 + Bx + C = 0 \) to have no real roots is that its discriminant must be less than zero:
\[ D = B^2 - 4AC < 0 \]
For our derivative:
- \( A = 3 \)
- \( B = 2a \)
- \( C = b \)

Thus, the discriminant is:
\[ (2a)^2 - 4(3)(b) < 0 \]
This simplifies to:
\[ 4a^2 - 12b < 0 \]
or
\[ a^2 < 3b \]

### Step 5: Analyze the given options
Now we need to check the given options for the ordered triplet (a, b, c) to see which satisfies the inequality \( a^2 < 3b \).

1. **Option 1: (2, 1, 3)**
- \( a^2 = 2^2 = 4 \)
- \( 3b = 3 \times 1 = 3 \)
- \( 4 < 3 \) is **false**.

2. **Option 2: (1, -1, 3)**
- \( a^2 = 1^2 = 1 \)
- \( 3b = 3 \times (-1) = -3 \)
- \( 1 < -3 \) is **false**.

3. **Option 3: (2, 2, 4)**
- \( a^2 = 2^2 = 4 \)
- \( 3b = 3 \times 2 = 6 \)
- \( 4 < 6 \) is **true**.

### Conclusion
The only ordered triplet (a, b, c) that satisfies the condition for monotonicity is:
\[ \text{(2, 2, 4)} \]