Use the input boxes below to define a function f(x) on an interval [a,b]. Use the checkboxes for a and b to exclude the endpoints from the domain of the function. Use the checkboxes to show/hide the critical points and local extrema in the interior of the interval (i.e., not at the endpoints). Use the "Trace y" button to trace points on the y-axis as you move the point c across the domain.

When we talk about extreme values of a function, we mean maximum or minimum values (without specifying which type). [br][br]Global (Absolute) Extreme Values: [br][list][*]f has a global maximum at x=c if [math]f(c)\ge f(x)[/math] for all x in the domain of f. [/*][*]f has a global minimum at x=c if [math]f(c)\le f(x)[/math] for all x in the domain of f. [/*][/list][br]Local (Relative) Extreme Values: [br][list][*]f has a local maximum at x=c if [math]f(c)\ge f(x)[/math] for all x in an interval containing x = c. [/*][*]f has a local minimum at x=c if [math]f(c)\le f(x)[/math] for all x in an interval containing x = c. [/*][/list][br]Critical Points:[br]The critical points of a function f are the points where either [math]f'(c)=0[/math] or [math]f'(c)[/math] does not exist. It turns out that all local extreme values must occur at a critical point. That is, the list of critical points is a list of potential extreme values. However, critical points do not always correspond to extreme values; so, the critical points must be tested to determine if there is an extreme value there.