Critical points can be found where the first derivative of a function is either equal to zero or it is undefined. They may indicate a trough, crest or rest stop and can be used to find the maxima or minima of a function.

[math]f\left(x\right)=x^3-3x+1[/math] [br][math]f'\left(x\right)=3x^2-3[/math] [br][math]f''\left(x\right)=6x[/math] J can [br][br]The function f(x)=x^3-3x+1 is pictured above along with both its first and second derivatives. Point J slides along the function of f(x). As it moves, the tangent line to the curve (k) moves with it. There is a black dotted line that runs through Point J, perpendicular to the tangent line to the curve. There are three colored, dashed vertical lines. These 3 lines have been added to mark where the behavior of the function[br][br]The critical points occur when the 1st derivative (the slope of the tangent line to the curve) is equal to zero, representing a trough (local minimum), crest (local maximum) or rest stop (a change in concavity.) Let's take a look at the first derivative to see what it can tell use about the function f(x).[br][br]f'(x)=[math]3x^2-3[/math][br]Where does f'(x)=0?[br] 0=[math]3x^2-3[/math][br] =[math]3\left(x^2-1\right)[/math][br] =(x-1)(x+1)[br] x=1[br] x=-1[br][br]This is verified visually when we look at f'(x) on the graph as it crosses the x axis at both x=1 and x=-1.[br]Slide Point J to the position of x=-1 on the graph of f(x). At this position, the slope of the tangent line to the curve is equal to 0, the black, dotted perpendicular line intersects f'(x) at a point that it crosses the x-axis. Is it a trough, a crest of a rest stop? Fortunately, the first derivative can help us determine the answer. In this example, the inflection point occurs where f(x) crosses the y-axis. [br][br]I hope that his applet helps you to visualize the relationship between the curve of a function and its first and second derivatives. [br]