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Introductory Calculus: Second Derivative Test
 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:
 
   
ExamplesExamples:
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.
 
    
ExamplesExamples:
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 .




M Ransom

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