PREVIOUS ANSWERS 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) \( \quad(x, y)=(\square) \)

To find the absolute extrema of the function \( f(x) = x^{4} - 2x^{3} + x + 1 \) on the interval \([-1, 3]\), it's essential to evaluate the function at critical points within the interval, as well as the endpoints. First, calculate the derivative \( f'(x) \) and set it to zero to find critical points. After determining the critical points, evaluate \( f(x) \) at those points and at the endpoints to identify both the absolute minimum and maximum values. Once you have found all the necessary function values, simply compare them to determine the minima and maxima, rounding to three decimal places. Good luck, and enjoy the graphing journey! One famous historical mathematician, Pierre de Fermat, contributed significantly to function theory and calculus long before they were formally established. His ideas on extrema, particularly found in his Method of Adequality, hinted at derivations that would shape calculus and optimization as we know them today. In real-world applications, optimization problems using functions like yours pop up everywhere! From economics, where businesses analyze profit maximization, to physics, where you might want to minimize energy, finding the absolute extrema can lead to critical decisions. Using graphing tools to visualize these problems can give fresh insights and clearer intuition into what the function is doing.