This applet was designed to help you intuitively (and informally) discover a very important concept in calculus. Definition: A CRITICAL NUMBER of a function f is any input value c in the domain of f for which either f'(c)=0 or f'(c) does not exist.
Directions:
There are graphs of 4 functions shown below. The names of these functions are c, h, f, and g.
You should note there is a point plotted on each function.
Move each of these points around to answer the following key question AND the questions that appear below this applet.
KEY QUESTION: Find all critical numbers of each function whose graph is given below.

Questions:
1) What is the minimum output of function c in the interval [-8, -4]? What is the maximum output of function c in the interval [-8, -4]?
2) At what input value(s) x does this/these maximum output(s) occur?
3) For each input value you listed in (2) above, find or estimate the value of c'(x) at each input value you listed.
4) What is the minimum output of function h in the interval [-7, 9]? What is the maximum output of function h in the interval [-7, 9]?
5) At what input value(s) x does this/these maximum output(s) occur?
6) For each input value you listed in (5) above, find or estimate the value of h'(x) at each input value you listed.
7) What is the minimum output of function f in the interval [-2, 2]? What is the maximum output of function f in the interval [-2, 2]?
8) At what input value(s) x does this/these maximum output(s) occur?
9) For each input value you listed in (8) above, find or estimate the value of f'(x) at each input value you listed.
10) What is the minimum output of function g in the interval [3, 6]? What is the maximum output of function g in the interval [3, 6]?
11) At what input value(s) x does this/these maximum output(s) occur?
12) For each input value you listed in (11) above, find or estimate the value of g'(x) at each input value you listed.
CONCLUDING QUESTION: Is there any relationship among your answers to questions (2), (5), (8), &/or (11) and your answer to the initial question that appeared above this applet? Explain as best as you can.