Inflection Points
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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 \(x > c_3 \).
Table Format For Inflection Points  

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.
Before we go on, try some practice problems.
Practice
Unless otherwise instructed, find the points of inflection and determine concavity for these functions.
\(f(x)=2x^3+6x^25x+1\)
Problem Statement
Calculate the inflection points of \(f(x)=2x^3+6x^25x+1\) and determine where the function is concave up or concave down.
Solution
video by Krista King Math 

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\(\displaystyle{f(x)=\frac{x^2+1}{x^2}}\)
Problem Statement
Calculate the inflection points of \(\displaystyle{f(x)=\frac{x^2+1}{x^2}}\) and determine where the function is concave up or concave down.
Solution
video by Krista King Math 

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\(f(x)=2+3x^2x^3\)
Problem Statement
Calculate the inflection points of \(f(x)=2+3x^2x^3\) and determine where the function is concave up or concave down.
Solution
video by PatrickJMT 

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\(h(x)=(x^21)^3\)
Problem Statement
Calculate the inflection points of \(h(x)=(x^21)^3\) and determine where the function is concave up or concave down.
Solution
video by PatrickJMT 

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\(f(x)=x^2 e^{4x}\)
Problem Statement
Calculate the inflection points of \(f(x)=x^2 e^{4x}\) and determine where the function is concave up or concave down.
Solution
video by MIP4U 

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Practice Instructions
Unless otherwise instructed, find the points of inflection and determine concavity for these functions.