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.) $$ f(x)=x^{4}-2 x^{3}+x+1,[-1,3] $$ minima (smaller $$ x $$-value) $$ \quad(x, y)=(1.5,0.813) $$ minima (larger $$ x $$-value) $$ \quad(x, y)=(0.25,1.238) $$

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To find the absolute extrema of the function $$ f(x) = x^{4} - 2x^{3} + x + 1 $$ on the interval $$[-1, 3]$$, we need to find the minimum and maximum values of the function within this interval.

Given:
- Minimum (smaller $$ x $$-value): $$(1.5, 0.813)$$
- Minimum (larger $$ x $$-value): $$(0.25, 1.238)$$

We can use a graphing utility to visualize the function and find the absolute extrema. However, since the specific graphing utility used is not provided, we will proceed with the given information to find the absolute extrema.

The absolute extrema of a function on a given interval are the minimum and maximum values of the function within that interval. The given information provides the minimum values of the function at $$ x = 1.5 $$ and $$ x = 0.25 $$ with corresponding $$ y $$-values of $$ 0.813 $$ and $$ 1.238 $$, respectively.

Therefore, the absolute extrema of the function $$ f(x) = x^{4} - 2x^{3} + x + 1 $$ on the interval $$[-1, 3]$$ are:
- Minimum (smaller $$ x $$-value): $$(1.5, 0.813)$$
- Minimum (larger $$ x $$-value): $$(0.25, 1.238)$$
The absolute extrema of the function $$ f(x) = x^{4} - 2x^{3} + x + 1 $$ on the interval $$[-1, 3]$$ are: - Minimum at $$ x = 1.5 $$ with $$ y = 0.813 $$ - Minimum at $$ x = 0.25 $$ with $$ y = 1.238 $$

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