The following graph shows the graph of [math]g(x)=sin^2(x)+1[/math], as you move the slider the point [math]A[/math] will move along. The graph of its derivative, [math]g'(x)[/math] can be shown/hidden by clicking the checkbox. What happens to the derivative when g(x) reaches a maximum/minimum?

If a function [math]f[/math] is continuous on the interval [math][a,b][/math], the function will have an absolute maximum and minimum on that interval, and these points can occur at the endpoints of the interval.[br][br]A good strategy to find the maximum/minimum of functions is:[br][list=1][*]Find the critical points of [math]f[/math] in [math][a,b][/math] ([math]c_1[/math], [math]c_2[/math], ..., [math]c_k[/math]), i.e. where the derivative is zero.[br][/*][*]Evaluate the function at the critical points [math]f(c_1)[/math],...[math]f(c_k)[/math],[math]f(a)[/math],[math]f(b)[/math].[br][/*][*]The highest/lowest value found this way will be the maximum/minimum of the function on [math][a,b][/math].[/*][/list]

On the next graph, try to identify the absolute maximum of [math]h(x)[/math] on [math][-1,2],[1,2][/math] and [math][-3,-2][/math]. What are the critical points of [math]h(x)=x^{\frac{2}{3}}[/math]? Is [math]h(x)[/math] defined at 0? What about [math]h'(x)[/math]?