Which function is positive for the entire interval [–3, –2]?

On a coordinate plane, a curved line with a minimum value of (0, negative 3) crosses the x-axis at (negative 3, 0) and (3, 0), and crosses the y-axis at (0, negative 3).

On a coordinate plane, a curved line with a minimum value of (2, negative 3) crosses the x-axis at (negative 1, 0) and (5, 0), and crosses the y-axis at (0, negative 1.5).

On a coordinate plane, a curved line with a minimum value of (2, 4) and a maximum value of (0.5, 6), crosses the x-axis at (negative 1.5, 0) and crosses the y-axis at (0, 5).

On a coordinate plane, a curved line with a minimum value of (negative 1.75, negative 3.9) and a maximum value of (0, 2), crosses the x-axis at (negative 2.2, 0), (negative 0.75, 0), and (0.75, 0), and crosses the y-axis at (0, 2).

To determine which function is positive for the entire interval \([-3, -2]\), we need to analyze each described function and its behavior within this interval.

1. **Function with minimum value at (0, -3)**:
   - Crosses the x-axis at \((-3, 0)\) and \((3, 0)\)
   - Crosses the y-axis at \((0, -3)\)

   This function crosses the x-axis at \((-3, 0)\), so it is not positive for the entire interval \([-3, -2]\).

2. **Function with minimum value at (2, -3)**:
   - Crosses the x-axis at \((-1, 0)\) and \((5, 0)\)
   - Crosses the y-axis at \((0, -1.5)\)

   This function has roots at \((-1, 0)\) and \((5, 0)\). Since \([-3, -2]\) is to the left of the root \((-1, 0)\), we need to determine the sign of the function in this interval. Since the minimum value at (2, -3) indicates the function is negative around \(x = 2\) and crosses the x-axis at \((-1, 0)\), it is negative for \([-3, -2]\).

3. **Function with minimum value at (2, 4)** and maximum value at (0.5, 6):
   - Crosses the x-axis at \((-1.5, 0)\)
   - Crosses the y-axis at \((0, 5)\)

   This function crosses the x-axis at \((-1.5, 0)\). Since \([-3, -2]\) is entirely to the left of this root and we know the function decreases towards this root, it is positive for \([-3, -2]\).

4. **Function with minimum value at (-1.75, -3.9)** and maximum value at (0, 2):
   - Crosses the x-axis at \((-2.2, 0)\), \((-0.75, 0)\), and \((0.75, 0)\)
   - Crosses the y-axis at \((0, 2)\)

   This function has a root at \((-2.2, 0)\). Since \([-3, -2]\) includes the root \((-2.2, 0)\), the function is not positive for the entire interval \([-3, -2]\).

Therefore, the function described in the third scenario is positive for the entire interval \([-3, -2]\).

Would you like details on any specific function or have any further questions?

Here are some more questions you might want to ask:
1. How to find the roots of a function?
2. What is the significance of the minimum and maximum values in a function?
3. How to determine the intervals where a function is positive or negative?
4. How to analyze the behavior of a function using its graph?
5. How do the coefficients of a polynomial function affect its graph?

**Tip:** When analyzing the behavior of functions, always check their critical points and where they cross the axes to understand their intervals of positivity or negativity.

Explore the analysis of functions to determine which function remains positive over the interval [-3, -2]. This problem involves understanding function behavior, interval analysis, and critical point examination. Suitable for college-level students studying calculus and function theory.

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