Question

A 2-row table with 9 columns. The first row is labeled x with entries negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, 4. The second row is labeled f of x with entries negative 54, negative 20, negative 4, 0, negative 2, negative 4, 0, 16, 50.
Which interval contains a local maximum for this function?



Which interval contains a local minimum for this function?

Answer 1

To determine which intervals contain local maxima and minima, we need to look for points where the function value \( f(x) \) changes direction from increasing to decreasing (for a local maximum) or from decreasing to increasing (for a local minimum).

### Identifying Intervals for Local Maxima:
A local maximum occurs where the function values increase up to a point and then decrease immediately after. Specifically:

1. At \( x = -1 \):
- \( f(-2) = -4 \rightarrow f(-1) = 0 \) (increase)
- \( f(-1) = 0 \rightarrow f(0) = -2 \) (decrease)
So, there is a local maximum at \( x = -1 \).

2. At \( x = 3 \):
- \( f(2) = 0 \rightarrow f(3) = 16 \) (increase)
- \( f(3) = 16 \rightarrow f(4) = 50 \) (increase)
Here, the values transition from increasing up to \( x = 3 \) and continue to increase beyond, thus \( x = 3 \) does not exhibit a local maximum based on given entries.

### Identifying Intervals for Local Minima:
A local minimum occurs where the function values decrease up to a point and then increase immediately after. Specifically:

1. At \( x = 1 \):
- \( f(0) = -2 \rightarrow f(1) = -4 \) (decrease)
- \( f(1) = -4 \rightarrow f(2) = 0 \) (increase)
So, there is a local minimum at \( x = 1 \).

### Summary:
- The interval that contains a **local maximum** is \((-2, 0)\), centered at \( x = -1 \).
- The interval that contains a **local minimum** is \((0, 2)\), centered at \( x = 1 \).