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.) \[ f(x)=x^{4}-2 x^{3}+x+1, \quad[-1,3] \] minima (smaller \( x \)-value) \( \quad(x, y)=(\square) \) minima (larger \( x \)-value) \( \quad(x, y)=(\square) \) maximum \[ (x, y)=(\square) \]

First, let's analyze the function \( f(x) = x^4 - 2x^3 + x + 1 \). To find the absolute extrema on the interval \([-1, 3]\), you would typically start by finding the derivative \( f'(x) \) and setting it equal to zero to locate critical points. You would then evaluate \( f(x) \) at these critical points as well as the endpoints of the interval, \( x = -1 \) and \( x = 3 \). Once you've plotted the function using a graphing utility, you would look for where the graph reaches its highest and lowest points within the specified interval. After performing these calculations, you might find the results: - minima (smaller \( x \)-value) \( (x, y) = (-1, 3) \) - minima (larger \( x \)-value) \( (x, y) = (1, 1) \) - maximum \( (x, y) = (3, 7) \) You might encounter important values and intervals for \( f(x) \) that help you understand its behavior throughout the specified range. Finding extrema can be part of a broader study in calculus, where understanding how functions behave is key to many applications, especially in fields like engineering and economics. Experimenting with different functions using graphing utilities like Desmos or GeoGebra enhances your intuition about function behavior and critical points visually!