If `f(x)=ax^(2)+bx+c, f(-1) gt (1)/(2), f(1) lt -1` and `f(-3)lt -(1)/(2)`, then

Options

A. `a=0`
B. `alt0`
C. `agt 0`
D. Sign of a can not be determined

Answer

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

### Step 1: Understand the function and conditions
The function is a quadratic function of the form \( f(x) = ax^2 + bx + c \). We are given three conditions:
1. \( f(-1) > \frac{1}{2} \)
2. \( f(1) < -1 \)
3. \( f(-3) < -\frac{1}{2} \)

### Step 2: Write the inequalities based on the conditions
We can express the conditions in terms of \( a \), \( b \), and \( c \):

1. For \( f(-1) > \frac{1}{2} \):
\[ a(-1)^2 + b(-1) + c > \frac{1}{2} \implies a - b + c > \frac{1}{2} \tag{1} \]

2. For \( f(1) < -1 \):
\[ a(1)^2 + b(1) + c < -1 \implies a + b + c < -1 \tag{2} \]

3. For \( f(-3) < -\frac{1}{2} \):
\[ a(-3)^2 + b(-3) + c < -\frac{1}{2} \implies 9a - 3b + c < -\frac{1}{2} \tag{3} \]

### Step 3: Analyze the inequalities
Now we have three inequalities to work with:

1. \( a - b + c > \frac{1}{2} \)
2. \( a + b + c < -1 \)
3. \( 9a - 3b + c < -\frac{1}{2} \)

### Step 4: Combine the inequalities
To find the relationship between \( a \), \( b \), and \( c \), we can manipulate these inequalities.

From (1) and (2):
- Subtract (2) from (1):
\[ (a - b + c) - (a + b + c) > \frac{1}{2} + 1 \]
This simplifies to:
\[ -2b > \frac{3}{2} \implies b < -\frac{3}{4} \tag{4} \]

Now, we can use (2) and (3):
- Subtract (2) from (3):
\[ (9a - 3b + c) - (a + b + c) < -\frac{1}{2} + 1 \]
This simplifies to:
\[ 8a - 4b < \frac{1}{2} \implies 16a - 8b < 1 \implies 2a - b < \frac{1}{8} \tag{5} \]

### Step 5: Determine the sign of \( a \)
To determine whether \( a \) is greater than, less than, or equal to zero, we look at the implications of the inequalities we've derived.

From (4), we have \( b < -\frac{3}{4} \). If we substitute this into (5):
\[ 2a - b < \frac{1}{8} \]
Since \( b \) is negative and less than \(-\frac{3}{4}\), this means \( -b \) is positive and greater than \(\frac{3}{4}\). Thus, \( 2a < \frac{1}{8} + b \) suggests that \( a \) must be negative to satisfy the inequalities, as \( b \) being negative will dominate the left side.

### Conclusion
Since the conditions lead us to conclude that \( a < 0 \), we can confidently state that:

\[ \text{The value of } a \text{ is less than } 0. \]