On a coordinate plane, a curved line with a minimum value of (negative 2, negative 6) and maximum values of (negative 3.75, 3) and (0.25, 13), crosses the x-axis at (negative 4, 0), (negative 3, 0), (negative 1, 0), and (1, 0), and crosses the y-axis at (0, 12).

To determine which interval contains the local minimum, we first identify the location of the local minimum specified in the problem. The local minimum point is indicated to be at the coordinates (-2, -6).

Next, we look at the intervals provided:

1. \([-1, 1]\)
2. \([1, 2]\)
3. \([-3, -1]\)
4. \([-5, -3]\)

Since the x-coordinate of the local minimum is -2, we need to find which interval it falls into:

- The interval \([-1, 1]\) does not include -2.
- The interval \([1, 2]\) does not include -2.
- The interval \([-3, -1]\) does not include -2.
- The interval \([-5, -3]\) does include -2 since -2 is between -5 and -3.

Therefore, the interval that contains the local minimum is:

\([-5, -3]\)