If `f(x) = x^(3) + a x^(2) + b x + c` has a maximum at x = -1 and minimum at x = 3. Find a + b.

Options

A. 12
B. `-12`
C. 9
D. 3

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the function \( f(x) = x^3 + ax^2 + bx + c \) has a maximum at \( x = -1 \) and a minimum at \( x = 3 \). We will use the first derivative test for maxima and minima.

### Step 1: Find the first derivative of \( f(x) \)
The first derivative \( f'(x) \) is given by:
\[ f'(x) = \frac{d}{dx}(x^3 + ax^2 + bx + c) = 3x^2 + 2ax + b \]

**Hint:** Remember that the derivative gives us the slope of the function, and we set it to zero to find critical points where maxima and minima occur.

### Step 2: Set the derivative to zero at the critical points
Since \( f(x) \) has a maximum at \( x = -1 \) and a minimum at \( x = 3 \), we set \( f'(-1) = 0 \) and \( f'(3) = 0 \).

1. For \( x = -1 \):
\[ f'(-1) = 3(-1)^2 + 2a(-1) + b = 3 - 2a + b = 0 \]
This simplifies to:
\[ -2a + b = -3 \quad \text{(Equation 1)} \]

2. For \( x = 3 \):
\[ f'(3) = 3(3)^2 + 2a(3) + b = 27 + 6a + b = 0 \]
This simplifies to:
\[ 6a + b = -27 \quad \text{(Equation 2)} \]

**Hint:** Setting the first derivative equal to zero at the critical points gives us a system of equations to solve for \( a \) and \( b \).

### Step 3: Solve the system of equations
We have the following two equations:
1. \( -2a + b = -3 \)
2. \( 6a + b = -27 \)

We can eliminate \( b \) by subtracting Equation 1 from Equation 2:
\[ (6a + b) - (-2a + b) = -27 - (-3) \]
This simplifies to:
\[ 6a + b + 2a - b = -27 + 3 \]
\[ 8a = -24 \]
\[ a = -3 \]

**Hint:** Use substitution or elimination to solve for one variable in terms of the other.

### Step 4: Substitute \( a \) back to find \( b \)
Now that we have \( a = -3 \), we can substitute this value back into Equation 1 to find \( b \):
\[ -2(-3) + b = -3 \]
\[ 6 + b = -3 \]
\[ b = -3 - 6 = -9 \]

**Hint:** Once you find one variable, substitute it back into one of the original equations to find the other variable.

### Step 5: Calculate \( a + b \)
Now we can find \( a + b \):
\[ a + b = -3 + (-9) = -12 \]

**Final Answer:**
\[ \boxed{-12} \]