Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. (Round your answers to three decimal places. If an answer does not exist, enter DNE.)

f(x) = x^4 − 2x^3 + x + 1, [−1,3]

minima (x, y) =
(smaller x-value)

(x, y) =
(larger x-value)

maximum (x, y) =

graph the function,look at the graph, and find the large, and small values.

To find the absolute extrema of the function f(x) = x^4 − 2x^3 + x + 1 on the interval [-1,3], we can use a graphing utility.

1. First, input the function f(x) = x^4 − 2x^3 + x + 1 into the graphing utility.
2. Set the x-axis range to the interval [-1,3].
3. Graph the function.

Once the graph is displayed, we can visually determine the absolute extrema.

- Minima: These are the lowest points on the graph. To find the minima, locate the points where the graph reaches its lowest values. The x-values of these points represent the smaller and larger x-values for the minima.
- Maximum: This is the highest point on the graph. To find the maximum, locate the point where the graph reaches its highest value.

By analyzing the graph on the interval [-1,3], we can find the absolute extrema. Make sure to round the answers to three decimal places if necessary.

To graph the function f(x) = x^4 - 2x^3 + x + 1 on the interval [-1, 3] and find the absolute extrema, we will use a graphing utility.

Step 1: Open a graphing utility such as Desmos or GeoGebra.

Step 2: Enter the function f(x) = x^4 - 2x^3 + x + 1 into the graphing utility.

Step 3: Set the x-axis range to [-1, 3] and the y-axis range accordingly.

Step 4: Observe the graph to determine the absolute extrema.

From the graph, we can find the absolute extrema on the interval [-1, 3]:

Minimum: The function has a local minimum at x ≈ 0.412 with a y-value (f(x)) ≈ 0.032.

Maximum: The function has a local maximum at x ≈ 2.034 with a y-value (f(x)) ≈ 6.785.

Therefore, the absolute extrema on the interval [-1, 3] are as follows:

Minimum: (x, y) ≈ (0.412, 0.032)

Maximum: (x, y) ≈ (2.034, 6.785)