`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 range of `f(x)` is (a) `[3,oo)` (b) `[-8,oo)` (c) `[-7,oo)` (d) `(-oo,6]`

Options

A. `[3,oo)`
B. `[-8,oo)`
C. `[-7,oo)`
D. `(-oo,6]`

Explanation

To find the range of the polynomial function \( y = f(x) \) given the conditions in the problem, we can follow these steps:

### Step 1: Analyze the given information
We know that:
- The function passes through the point (0, 1), so \( f(0) = 1 \).
- The function is increasing in the intervals \( (1, 2) \) and \( (3, \infty) \).
- The function is decreasing in the intervals \( (-\infty, 1) \) and \( (2, 3) \).
- We also know that \( f(1) = -8 \).

### Step 2: Identify critical points
From the increasing and decreasing intervals, we can identify critical points:
- At \( x = 1 \), the function changes from decreasing to increasing, indicating that \( x = 1 \) is a local minimum.
- At \( x = 2 \), the function changes from increasing to decreasing, indicating that \( x = 2 \) is a local maximum.
- At \( x = 3 \), the function changes from decreasing to increasing, indicating that \( x = 3 \) is another local minimum.

### Step 3: Determine the values at critical points
We have:
- \( f(1) = -8 \) (local minimum)
- Since the function is increasing after \( x = 1 \) until \( x = 2 \), we need to find \( f(2) \) which will be the maximum value.
- After \( x = 2 \), the function decreases until \( x = 3 \) and then increases again.

### Step 4: Analyze the behavior of the function
As \( x \to -\infty \), since the function is decreasing, \( f(x) \) will approach \( -\infty \).
As \( x \to \infty \), since the function is increasing, \( f(x) \) will approach \( \infty \).

### Step 5: Find the maximum value at \( x = 2 \)
Since we don't have the exact polynomial, we can infer that the maximum value at \( x = 2 \) must be greater than \( -8 \) because the function increases from \( -8 \) at \( x = 1 \) to some maximum value at \( x = 2 \).

### Step 6: Determine the range
The minimum value of the function is \( -8 \) at \( x = 1 \), and since the function increases to some maximum value at \( x = 2 \) and then decreases again, the range of \( f(x) \) will start from the minimum value \( -8 \) and go to \( \infty \).

Thus, the range of the function \( f(x) \) is:
\[ [-8, \infty) \]

### Conclusion
The correct option is (b) \([-8, \infty)\).