`y= f(x)` is a polynomial function passing through point (0, 1) and which increases in the intervals `(1, 2) and (3, oo)` and decreases in the intervals `(oo,1) and (2, 3).`
If `f(1) = -8,` then the value of `f(2)` is

Options

A. -3
B. -6
C. -20
D. -7

Explanation

To solve the problem, we need to find the polynomial function \( f(x) \) that meets the given conditions and then evaluate \( f(2) \).

### Step 1: Determine the derivative of the polynomial
Given that the polynomial \( f(x) \) increases in the intervals \( (1, 2) \) and \( (3, \infty) \) and decreases in the intervals \( (-\infty, 1) \) and \( (2, 3) \), we can conclude that the critical points (where the derivative is zero) are at \( x = 1, 2, \) and \( 3 \). Therefore, we can express the derivative as:
\[ f'(x) = a(x - 1)(x - 2)(x - 3) \]
where \( a \) is a constant.

### Step 2: Expand the derivative
Expanding \( f'(x) \):
\[ f'(x) = a(x^3 - 6x^2 + 11x - 6) \]

### Step 3: Integrate to find \( f(x) \)
Integrating \( f'(x) \):
\[ f(x) = \int f'(x) \, dx = a\left(\frac{x^4}{4} - 2x^3 + \frac{11x^2}{2} - 6x\right) + C \]
where \( C \) is the constant of integration.

### Step 4: Use the point (0, 1) to find \( C \)
Since the polynomial passes through the point \( (0, 1) \):
\[ f(0) = a\left(0\right) + C = 1 \implies C = 1 \]
Thus, we have:
\[ f(x) = a\left(\frac{x^4}{4} - 2x^3 + \frac{11x^2}{2} - 6x\right) + 1 \]

### Step 5: Use the condition \( f(1) = -8 \) to find \( a \)
Substituting \( x = 1 \):
\[ f(1) = a\left(\frac{1}{4} - 2 + \frac{11}{2} - 6\right) + 1 = -8 \]
Calculating inside the parentheses:
\[ \frac{1}{4} - 2 + \frac{11}{2} - 6 = \frac{1}{4} - \frac{8}{4} + \frac{22}{4} - \frac{24}{4} = \frac{1 - 8 + 22 - 24}{4} = \frac{-9}{4} \]
Thus, we have:
\[ a\left(\frac{-9}{4}\right) + 1 = -8 \]
This simplifies to:
\[ -\frac{9a}{4} + 1 = -8 \implies -\frac{9a}{4} = -9 \implies a = 4 \]

### Step 6: Write the final form of \( f(x) \)
Substituting \( a = 4 \) back into \( f(x) \):
\[ f(x) = 4\left(\frac{x^4}{4} - 2x^3 + \frac{11x^2}{2} - 6x\right) + 1 \]
This simplifies to:
\[ f(x) = x^4 - 8x^3 + 22x^2 - 24x + 1 \]

### Step 7: Calculate \( f(2) \)
Now we evaluate \( f(2) \):
\[ f(2) = 2^4 - 8(2^3) + 22(2^2) - 24(2) + 1 \]
Calculating each term:
\[ = 16 - 8(8) + 22(4) - 48 + 1 \]
\[ = 16 - 64 + 88 - 48 + 1 \]
Combining these:
\[ = 16 - 64 + 88 - 48 + 1 = -7 \]

Thus, the value of \( f(2) \) is:
\[ \boxed{-7} \]