[list][*]Use the input box to define the function y = f(x). Use the input boxes for a and b to define the endpoints of the domain of f. Use the checkboxes to include/exclude the endpoints. [/*][*]Use the checkbox for Critical Points or Interior Extrema to show/hide critical points or interior extreme values, respectively. [/*][*]Use the checkbox for Monotonicity to highlight the section of the graph corresponding to the location of the point c. [/*][*]Use the Derivative checkbox to show the graph of the derivative function. [/*][/list]

Our goal is to be able to describe [b]transitions [/b]in the [b]behavior [/b]of a function (e.g., from increasing to decreasing, from concave up to concave down). These changes in a function correspond to easier-to-find changes in its derivative(s). Because the [b]monotonicity [/b](increasing/decreasing behavior) of a function is tied to the [b]sign [/b](positive/negative) of its derivative, describing the monotonicity of f requires finding where the derivative f' is positive, negative, or zero. [br][br]Assuming certain nice properties of the derivative function (i.e., that it is continuous), the derivative could not change from positive to negative (or vice versa) without passing through a value of 0. A[b] critical point[/b] is a point where the derivative is equal to zero (or does not exist). Hence, a critical point is a location where the function [i]has the potential to change monotonicity[/i]. [br][br]The illustration above demonstrates that the critical points of a function give [i]potential candidates[/i] for [b]extreme values[/b] (maximum/minimum values). [i]While all extreme values occur at critical points, not every critical point results in an extreme value. So, we have to [b]test [/b]critical points to determine whether there is a maximum or minimum value there. [/i]