J can
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
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).
f'(x)=
Where does f'(x)=0?
0=
=
=(x-1)(x+1)
x=1
x=-1
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.
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.
I hope that his applet helps you to visualize the relationship between the curve of a function and its first and second derivatives.