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Using The Second Derivative For Graphing 

On this page we discuss using the second derivative to help determine what the graph of a function looks like. The techniques discussed are concavity, inflection points and the second derivative test. 
Concavity 
Concavity relates to how the graph is curving, either upward or downward. If a graph is curving upward, then it looks like a cup and it could hold water. The graph of \(y=x^2\) is concave upward.
If the graph is curving downward, then it looks like an arched roof and could keep you dry you in the rain. The graph of \(y=x^2\) is concave downward.
This plot shows several examples of concavity to help you get a feel for what they look like.
The concept of concavity parallels increasing and decreasing intervals. We learned that increasing and decreasing sections of a function change only at critical points. Similarly, concavity is related to the second derivative and changes only at inflection points.
Here is a quick video clip explaining this idea again.
PatrickJMT  Concavity  
Inflection Points 
We use inflection points to help us determine the where concavity changes. Basically, concavity will change only at inflection points. To find inflection points, we use a similar procedure as we did for critical points, except we us the second derivative. So we start by taking the derivative twice, set the result to zero and solve for the xvalues. We also look at values where the second derivative is not defined but the points are in the domain of the original function.
The inflection points allow us to determine concavity. We can use the following table format to organize the information. We assume here that we have a function \(g(x)\) with break points at \(x=c_1\), \(x=c_2\) and \(x=c_3\) and the function is defined for \(x < c_1\) and for \(x > c_3 \).
Interval 
\( \infty < x < c_1 \) 
\( c_1 < x < c_2 \) 
\( c_2 < x < c_3 \) 
\( c_3 < x < \infty \) 

Test xvalue 

Sign of \(g''(x)\) 

Conclusion 
Possible conclusions include concave upward or concave downward.
Note: The break points include points of inflection and discontinuities. Basically, the entire domain needs to be covered by the intervals in the first row.
The test values can be any point in the open interval in each column.
Second Derivative Test 
The second derivative test is interesting in that it tests for the same information as the first derivative test. On the good side it is easier to use. On the bad side, it doesn't always work.
The idea is that you find the second derivative and then plug the critical points in the second derivative. If the result is less than zero, then you have a relative maximum, greater than zero, a relative minimum. If you get zero, the test is inconclusive and you drop back and use the first derivative test. To organize your information, you can use a table in this format.
Interval/Point 
\( x = c_1 \) 
\( x = c_2 \) 
\( x = c_3 \) 

Test xvalue 

Sign of \(g''(x)\) 

Conclusion 
The possible conclusions in the last row above maximum, minimum or inconclusive. In the last case, we need to use the first derivative test.
Before working practice problems, take a few minutes to watch this quick video clip explaining the second derivative test in more detail.
PatrickJMT  Second Derivative Test  
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Practice Problems 

Instructions   Unless otherwise instructed, determine the concavity, inflection points or maximums/minimums using the second derivative test of these functions, giving your answers in exact terms.
Level A  Basic 
Practice A01  

concavity, inflection  
solution 
Practice A02  

concavity, inflection  
solution 
Practice A03  

concavity, inflection  
solution 
Practice A04  

maximums/minimums  
solution 
Practice A05  

Suppose \(f(x)\) has a critical point at \(x=4\) and \(f''(4) = 3\). What can be said about \(f(x)\) at \(x=4\)?  
solution 
Level B  Intermediate 
Practice B01  

concavity, inflection  
solution 
Practice B02  

maximums/minimums  
solution 