If `f(x)=ax^(2)+bx+c` and `f(-1) ge -4`, `f(1) le 0` and `f(3) ge 5`, then the least value of `a` is

Options

A. `1//4`
B. `1//8`
C. `1//3`
D. `-1//3`

Explanation

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

### Step 1: Set up the inequalities based on the conditions
We have three conditions based on the values of \( f(x) \):

1. \( f(-1) \geq -4 \)
2. \( f(1) \leq 0 \)
3. \( f(3) \geq 5 \)

### Step 2: Write the inequalities
Using the function \( f(x) = ax^2 + bx + c \), we can express these conditions as:

1. For \( f(-1) \):
\[ f(-1) = a(-1)^2 + b(-1) + c = a - b + c \geq -4 \quad \text{(Inequality 1)} \]
Which simplifies to:
\[ a - b + c + 4 \geq 0 \quad \text{(1)} \]

2. For \( f(1) \):
\[ f(1) = a(1)^2 + b(1) + c = a + b + c \leq 0 \quad \text{(Inequality 2)} \]
Which simplifies to:
\[ a + b + c \leq 0 \quad \text{(2)} \]

3. For \( f(3) \):
\[ f(3) = a(3)^2 + b(3) + c = 9a + 3b + c \geq 5 \quad \text{(Inequality 3)} \]
Which simplifies to:
\[ 9a + 3b + c - 5 \geq 0 \quad \text{(3)} \]

### Step 3: Rearranging the inequalities
Now we can rearrange these inequalities for easier manipulation:

1. From (1):
\[ a - b + c \geq -4 \quad \Rightarrow \quad a - b + c \geq -4 \quad \text{(1)} \]

2. From (2):
\[ a + b + c \leq 0 \quad \Rightarrow \quad -a - b - c \geq 0 \quad \text{(2)} \]

3. From (3):
\[ 9a + 3b + c \geq 5 \quad \Rightarrow \quad 9a + 3b + c - 5 \geq 0 \quad \text{(3)} \]

### Step 4: Combine the inequalities
Now we will add (1) and (2):
\[ (a - b + c) + (-a - b - c) \geq -4 + 0 \]
This simplifies to:
\[ -2b \geq -4 \quad \Rightarrow \quad b \leq 2 \quad \text{(4)} \]

Next, we will add (2) and (3):
\[ (a + b + c) + (9a + 3b + c) \geq 0 + 5 \]
This simplifies to:
\[ 10a + 4b + 2c \geq 5 \quad \text{(5)} \]

### Step 5: Solve for \( a \)
Now we have the inequalities:
1. \( b \leq 2 \) from (4)
2. We can use (2) to express \( c \):
\[ c = -a - b \quad \text{(from (2))} \]
Substitute \( c \) in (5):
\[ 10a + 4b + 2(-a - b) \geq 5 \]
Simplifying gives:
\[ 10a + 4b - 2a - 2b \geq 5 \]
\[ 8a + 2b \geq 5 \]
\[ 4a + b \geq \frac{5}{2} \quad \text{(6)} \]

### Step 6: Substitute \( b \) from (4)
Substituting \( b = 2 \) into (6):
\[ 4a + 2 \geq \frac{5}{2} \]
\[ 4a \geq \frac{5}{2} - 2 \]
\[ 4a \geq \frac{1}{2} \]
\[ a \geq \frac{1}{8} \]

### Conclusion
Thus, the least value of \( a \) is:
\[ \boxed{\frac{1}{8}} \]