Supremum Pada Masalah Optimasi Fungsi Real Berdimensi Satu Menggunakan Python

Abstract

The concept of supremum is fundamental in real analysis and plays a crucial role in the optimization of single-variable real functions. In practice, not all functions attain their supremum explicitly, which necessitates numerical approaches to evaluate their behavior computationally. This study aims to analyze the supremum of several one-dimensional real functions with different characteristics using a grid-search method implemented in Python. Four functions were examined: a parabolic function, a rational function with a sharp peak, a discontinuous piecewise function, and a function with a vertical asymptote. The analysis involved modeling the functions, discretizing the domain, performing numerical approximation of the supremum, verifying the results against analytical values, and using graphical visualization to observe the function behavior near the supremum. The findings indicate that the supremum of the parabolic, rational, and piecewise functions can be accurately identified, with results consistent with analytical expectations despite minor deviations caused by grid resolution limitations in the rational function. Meanwhile, the function with a vertical asymptote yields an unbounded supremum, which cannot be attained within the domain. These results demonstrate that Python provides stable and reliable numerical estimates of the supremum across various types of one-dimensional real functions, validating the effectiveness of computational methods in supporting conceptual understanding of supremum.