To determine whether the function \( f: [0, 3] \to [1, 29] \) defined by \( f(x) = 2x^3 - 15x^2 + 36x + 1 \) is one-to-one (1-1) or onto (on2), we will follow these steps:
### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first differentiate it:
\[
f'(x) = \frac{d}{dx}(2x^3 - 15x^2 + 36x + 1)
\]
Using the power rule, we get:
\[
f'(x) = 6x^2 - 30x + 36
\]
### Step 2: Factor the derivative
Next, we can factor the derivative to find the critical points:
\[
f'(x) = 6(x^2 - 5x + 6) = 6(x-2)(x-3)
\]
### Step 3: Determine the critical points
The critical points occur when \( f'(x) = 0 \):
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
### Step 4: Analyze the sign of the derivative
We will analyze the sign of \( f'(x) \) in the intervals determined by the critical points \( x = 2 \) and \( x = 3 \):
- For \( x < 2 \) (e.g., \( x = 0 \)):
\[
f'(0) = 6(0-2)(0-3) = 36 > 0 \quad \text{(increasing)}
\]
- For \( 2 < x < 3 \) (e.g., \( x = 2.5 \)):
\[
f'(2.5) = 6(2.5-2)(2.5-3) = 6(0.5)(-0.5) = -1.5 < 0 \quad \text{(decreasing)}
\]
- For \( x > 3 \) (e.g., \( x = 4 \)):
\[
f'(4) = 6(4-2)(4-3) = 12 > 0 \quad \text{(increasing)}
\]
### Step 5: Determine the behavior of the function
From the analysis:
- The function is increasing on \( [0, 2] \).
- The function is decreasing on \( [2, 3] \).
### Step 6: Evaluate the function at the endpoints and critical points
Now we will evaluate \( f(x) \) at the endpoints and critical points:
1. \( f(0) = 2(0)^3 - 15(0)^2 + 36(0) + 1 = 1 \)
2. \( f(2) = 2(2)^3 - 15(2)^2 + 36(2) + 1 = 16 - 60 + 72 + 1 = 29 \)
3. \( f(3) = 2(3)^3 - 15(3)^2 + 36(3) + 1 = 54 - 135 + 108 + 1 = 28 \)
### Step 7: Determine the range of the function
From the evaluations:
- Minimum value: \( f(0) = 1 \)
- Maximum value: \( f(2) = 29 \)
- Value at \( f(3) = 28 \)
Thus, the range of \( f(x) \) over the interval \( [0, 3] \) is \( [1, 29] \).
### Step 8: Conclusion
Since the function is increasing on \( [0, 2] \) and decreasing on \( [2, 3] \), it is not one-to-one (1-1) because it does not pass the horizontal line test. However, since the range matches the codomain \( [1, 29] \), the function is onto (on2).
### Final Answer
The function \( f(x) = 2x^3 - 15x^2 + 36x + 1 \) is onto (on2) but not one-to-one (1-1).
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