If the parabola `y=ax^2+bx+c` has vertex at(4,2)and ` a in [1,3]` then the difference beteween the extreme value of abc is equal to

Options

A. 3600
B. 144
C. 3456
D. none of these

Explanation

To solve the problem step by step, we will analyze the given parabola and its vertex, derive the relationships between the coefficients \(a\), \(b\), and \(c\), and then find the extreme values of the product \(abc\).

### Step 1: Identify the vertex form of the parabola
The vertex of the parabola given by the equation \(y = ax^2 + bx + c\) is at the point \((h, k)\), where \(h = -\frac{b}{2a}\) and \(k = f(h)\). Given that the vertex is at \((4, 2)\), we can set up the following equations:
\[ -\frac{b}{2a} = 4 \quad \text{(1)} \]
\[ f(4) = 2 \quad \text{(2)} \]

### Step 2: Solve for \(b\) using equation (1)
From equation (1):
\[ b = -8a \]

### Step 3: Substitute \(b\) into equation (2)
Substituting \(b = -8a\) into the equation for \(f(4)\):
\[ f(4) = a(4^2) + b(4) + c = 2 \]
This simplifies to:
\[ 16a - 32a + c = 2 \]
\[ -16a + c = 2 \quad \text{(3)} \]
From equation (3), we can express \(c\) in terms of \(a\):
\[ c = 16a + 2 \]

### Step 4: Find the product \(abc\)
Now we have:
- \(a = a\)
- \(b = -8a\)
- \(c = 16a + 2\)

The product \(abc\) can be expressed as:
\[ abc = a \cdot (-8a) \cdot (16a + 2) \]
\[ abc = -8a^2(16a + 2) = -128a^3 - 16a^2 \]

### Step 5: Differentiate \(abc\) with respect to \(a\)
To find the extreme values, we differentiate \(abc\) with respect to \(a\):
\[ \frac{d(abc)}{da} = -384a^2 - 32a \]
Setting the derivative to zero to find critical points:
\[ -384a^2 - 32a = 0 \]
Factoring out \(-16a\):
\[ -16a(24a + 2) = 0 \]
This gives us:
\[ a = 0 \quad \text{or} \quad a = -\frac{1}{12} \]
However, since \(a\) must be in the range \([1, 3]\), we will evaluate \(abc\) at the endpoints of the interval.

### Step 6: Evaluate \(abc\) at the endpoints \(a = 1\) and \(a = 3\)
1. For \(a = 1\):
\[ b = -8(1) = -8 \]
\[ c = 16(1) + 2 = 18 \]
\[ abc = 1 \cdot (-8) \cdot 18 = -144 \]

2. For \(a = 3\):
\[ b = -8(3) = -24 \]
\[ c = 16(3) + 2 = 50 \]
\[ abc = 3 \cdot (-24) \cdot 50 = -3600 \]

### Step 7: Find the difference between the extreme values
The extreme values of \(abc\) are \(-144\) and \(-3600\). The difference is:
\[ \text{Difference} = -144 - (-3600) = -144 + 3600 = 3456 \]

### Final Answer
The difference between the extreme values of \(abc\) is \(3456\).