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  2. Extreme value theorem - Wikipedia

    en.wikipedia.org/wiki/Extreme_value_theorem

    The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.

  3. Golden-section search - Wikipedia

    en.wikipedia.org/wiki/Golden-section_search

    The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.

  4. Maximum and minimum - Wikipedia

    en.wikipedia.org/wiki/Maximum_and_minimum

    x −x: Unique global maximum over the positive real numbers at x = 1/e. x 3 /3 x: First derivative x 2 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum.

  5. Derivative test - Wikipedia

    en.wikipedia.org/wiki/Derivative_test

    Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose further that f is continuous at x. If there exists a positive number r > 0 such that f is weakly increasing on (x r, x] and weakly decreasing on [x, x + r), then f has a local maximum at x.

  6. Fermat's theorem (stationary points) - Wikipedia

    en.wikipedia.org/wiki/Fermat's_theorem...

    Assume that function f has a maximum at x 0, the reasoning being similar for a function minimum. If x 0 ∈ ( a , b ) {\displaystyle \displaystyle x_{0}\in (a,b)} is a local maximum then, roughly, there is a (possibly small) neighborhood of x 0 {\displaystyle \displaystyle x_{0}} such as the function "is increasing before" and "decreasing after ...

  7. Global optimization - Wikipedia

    en.wikipedia.org/wiki/Global_optimization

    Global optimization is distinguished from local optimization by its focus on finding the minimum or maximum over the given set, as opposed to finding local minima or maxima. Finding an arbitrary local minimum is relatively straightforward by using classical local optimization methods. Finding the global minimum of a function is far more ...

  8. When are College Football Playoff rankings released? What to know

    www.aol.com/college-football-playoff-rankings...

    Byes: No. 1 Oregon, No. 2 Ohio State, No. 3 Georgia, No. 4 BYU First round : No. 12 Notre Dame at No. 5 Miami First round : No. 11 SMU Boise State at No. 6 Tennessee

  9. Bisection method - Wikipedia

    en.wikipedia.org/wiki/Bisection_method

    Examine the sign of f(c) and replace either (a, f(a)) or (b, f(b)) with (c, f(c)) so that there is a zero crossing within the new interval. When implementing the method on a computer, there can be problems with finite precision, so there are often additional convergence tests or limits to the number of iterations.