Question

A graph of a fifth degree polynominal function approaches infinity as x approaches negative infinity and approaches negative infinity as x approaches infinity.The function starts in quadrant 2 and decreases to point left-parenthesis negative 4 comma negative 2 right-parenthesis. It then increases to point left-parenthesis negative 1 comma 2 right-parenthesis and decreases back down to left-parenthesis 2 comma 0 right-parenthesis. From here it increases again to point left-parenthesis 4 comma 3 right-parenthesis and then decreases infinitely.
Question
Use the graph to answer the question.
What is the maximum value for the function on the interval left bracket negative 2 comma 3 right bracket?
(1 point)
Responses

negative 1
Image with alt text: negative 1

2
2

3
3

4

Answer 1

To determine the maximum value of the function on the interval \([-2, 3]\), we should look at the values of the function at the endpoints and any critical points within the interval.

From the information provided:

- The function decreases to \((-4, -2)\) – we are not interested in this point because it is outside the interval.
- It then increases to \((-1, 2)\), which is a local maximum.
- It decreases to \((2, 0)\), and then it increases to \((4, 3)\) – however, we are only focused on the interval \([-2, 3]\).

The critical points within \([-2, 3]\) appear to be:

1. \(f(-1) = 2\) (local maximum)
2. \(f(2) = 0\) (local minimum)
3. For \(x = -2\) (the left endpoint), we need to identify the function's value.

Since you're looking for the maximum value over the interval \([-2, 3]\), you need to evaluate the values:

- \(f(-2)\) (which was not given)
- \(f(-1) = 2\)
- \(f(2) = 0\)
- We don't consider \(f(3)\) explicitly, but since the function is said to decrease as x approaches positive infinity, it would be less than or equal to \(f(4) = 3\).

Since \(f(-1) = 2\) is given and appears to be the highest among identified points, we can conclude:

The maximum value for the function on the interval \([-2, 3]\) is:
**2**