AlgebraLab

Introductory Calculus: Second Derivative Test

By M Ransom

 Regions of Concavity up or down AND Maximum and Minimum Values of a Function
 
We look first at an example function . We calculate the derivative:
 
 
Notice that the derivative equals 0 when x = -1 or +1.
 
It can be shown that the derivative is positive when x < -1 and x > +1, and is negative when -1 < x < +1.
 
Next we calculate the second derivative:
 
.
 
Notice that the second derivative is 0 when x = 0.
 
It can be shown that the second derivative is negative for x < 0 and positive for x > 0.
 
Below, we show the graph of  twice. In the second graph, tangent lines are drawn which are horizontal (reminder: lines which are horizontal have a slope of 0). Some observations about these diagrams are made below the graphs.
 
  1. appears to be “curving downward” until x = 0.
  2. appears to be “curving upward” from x = 0 onward.
  3. has a relative maximum at (-1, 2) where is “curving downward” and a minimum at (1, -2) where is “curving upward.”
  4. changes from concave down to concave up at the point (0, 0) and the second derivative is 0.
 
These observations can be summarized as follows:
 
  1. is concave down when x < 0
         (notice the second derivative is negative).
  2. is concave up when 0 < x
         (notice the second derivative is positive).
  3. has a relative maximum at (-1, 2) and .

  4. has a relative minimum at (1, -2) and .

     
 
Generalized Results

  1. A function is concave up at a point if there is an interval containing c and the second derivative is positive on this interval. If , there is a relative minimum at .
  2. A function is concave down at a point if there is an interval containing c and the second derivative is negative on this interval. If , there is a relative maximum at .
  3. A function possibly has a point of inflection at a point where the second derivative is exactly 0 (or as we shall see, at a point where the second derivative does not exist).
 
 
 
 
Use the following information about  to answer the next three questions:
 
   
Put the information about the first and second derivatives into sign charts.
 
 
Sketch a possible graph of paying close attention to the points given, the regions of increase and decrease, and the regions of differing concavity.
 

At what value(s) of is there a relative min or max? How do you know?

 
 
 
Use the following information about  to answer the next two questions.
 
    
Find the critical points of from the first derivative. Use these to construct a sign chart for .
 
Find the critical points of from the second derivative. Use these to construct a sign chart for .