Let `f(x) =ax^(2) + bx + c` and `f(-1) lt 1, f(1) gt -1, f(3) lt -4` and `a ne 0`, then

Options

A. (a) a>0
B. (b) a<0
C. (c) Sign of a can't be determined
D. (a) none of the above

Explanation

To solve the problem step by step, we will analyze the function \( f(x) = ax^2 + bx + c \) using the given conditions.

### Step 1: Write down the conditions from the problem
We have three conditions based on the function evaluated at specific points:
1. \( f(-1) < 1 \)
2. \( f(1) > -1 \)
3. \( f(3) < -4 \)

### Step 2: Express each condition in terms of \( a \), \( b \), and \( c \)

1. For \( f(-1) < 1 \):
\[ f(-1) = a(-1)^2 + b(-1) + c = a - b + c < 1 \]
This can be rearranged to:
\[ a - b + c < 1 \quad \text{(Equation 1)} \]

2. For \( f(1) > -1 \):
\[ f(1) = a(1)^2 + b(1) + c = a + b + c > -1 \]
This can be rearranged to:
\[ a + b + c > -1 \quad \text{(Equation 2)} \]

3. For \( f(3) < -4 \):
\[ f(3) = a(3)^2 + b(3) + c = 9a + 3b + c < -4 \]
This can be rearranged to:
\[ 9a + 3b + c < -4 \quad \text{(Equation 3)} \]

### Step 3: Analyze the inequalities

Now we have three inequalities:
1. \( a - b + c < 1 \)
2. \( a + b + c > -1 \)
3. \( 9a + 3b + c < -4 \)

### Step 4: Combine the inequalities

Let's add Equation 1 and Equation 3:
\[ (a - b + c) + (9a + 3b + c) < 1 - 4 \]
This simplifies to:
\[ 10a + 2b + 2c < -3 \]
Dividing the entire inequality by 2 gives:
\[ 5a + b + c < -\frac{3}{2} \quad \text{(Equation 4)} \]

Now, we will add Equation 2 and Equation 4:
\[ (a + b + c) + (5a + b + c) > -1 - \frac{3}{2} \]
This simplifies to:
\[ 6a + 2b + 2c > -\frac{5}{2} \]
Dividing the entire inequality by 2 gives:
\[ 3a + b + c > -\frac{5}{4} \quad \text{(Equation 5)} \]

### Step 5: Analyze the results

Now we have two new inequalities:
1. \( 5a + b + c < -\frac{3}{2} \) (Equation 4)
2. \( 3a + b + c > -\frac{5}{4} \) (Equation 5)

### Step 6: Subtract Equation 5 from Equation 4

Subtract Equation 5 from Equation 4:
\[ (5a + b + c) - (3a + b + c) < -\frac{3}{2} + \frac{5}{4} \]
This simplifies to:
\[ 2a < -\frac{3}{2} + \frac{5}{4} \]
Finding a common denominator (4):
\[ 2a < -\frac{6}{4} + \frac{5}{4} = -\frac{1}{4} \]
Dividing both sides by 2 gives:
\[ a < -\frac{1}{8} \]

### Conclusion

Since \( a < -\frac{1}{8} \), we conclude that \( a < 0 \).

Thus, the answer is:
**Option B: \( a < 0 \)**.