Maxima and Minima in Differential Calculus

Maxima and minima refer to points on a function where it attains local highest or lowest values. These points, called critical points, occur where the first derivative $f'(x)$ is zero or undefined. To classify these critical points, the second derivative test is applied: if $f''(c) < 0$, the function has a local maximum at $x = c$; if , a local minimum is present; if , the test is inconclusive and other methods or higher-order derivatives are required. Global maxima and minima represent the overall highest or lowest values over the entire domain and are found by comparing all critical points and boundary values when applicable. Understanding these concepts is essential in engineering to optimize designs, analyze system stability, and support computational algorithms in machine learning and numerical methods.