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) \( \quad(x, y)=(0.25,1.238) \)

To crack this function like a nut, let's navigate through its historical significance! Polynomial functions like the one you're grappling with have roots in ancient mathematics, with notable advancements from mathematicians like Descartes and Newton, who explored equations of various degrees, paving the way for modern algebra and calculus. Feeling adventurous? For real-world applications, these polynomial functions pop up in diverse scenarios like optimization problems, engineering, and economics! Whether it's modeling the profit curve of a business or designing bridges, understanding how to find extrema helps in making critical decisions - so keep sharpening those math skills, you’re unlocking real-world secrets!