maxima

Maxima is the plural of maximum and refers to the greatest value that a function, sequence, or set attains. A maximum can be global (attained over the entire domain) or local (the greatest value within a neighborhood). If a function is continuous on a closed and bounded domain, the extreme value theorem guarantees that both a maximum and a minimum exist; however, the maximum may not be unique.

Critical points: For differentiable functions, local maxima occur where the derivative is zero or undefined. The second derivative test (in one variable) or the Hessian matrix (in several variables) helps determine whether a critical point is a local maximum, a local minimum, or a saddle point. In constrained optimization, maxima can be found using methods such as Lagrange multipliers, which incorporate constraints into the optimization process.

Examples: The function f(x) = −(x − 2)² + 3 has a global maximum of 3 at x = 2. A function defined on a closed interval may attain its maximum either at interior critical points or at the endpoints of the interval.

Numerical methods: When an explicit solution is difficult, algorithms such as gradient ascent, Newton’s method for optimization, or other global optimization techniques can locate maxima, sometimes with guarantees under specific conditions.

Maxima (software): Maxima is a free, open-source computer algebra system derived from Macsyma. It performs symbolic computation such as algebra, differentiation, integration, solving equations, series, and plotting. It runs on various operating systems and is accessible through multiple user interfaces, including wxMaxima.