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17Calculus Derivatives - Inflection Points

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Inflection Points

Topics You Need To Understand For This Page

derivatives graphing 2nd derivative 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 x-values. 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 x-value

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.

Math Word Problems Demystified

Practice

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

\(f(x)=2x^3+6x^2-5x+1\)

Problem Statement

Calculate the inflection points of \(f(x)=2x^3+6x^2-5x+1\) and determine where the function is concave up or concave down.

Solution

Krista King Math - 1349 video 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

Krista King Math - 1350 video solution

video by Krista King Math

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\(f(x)=2+3x^2-x^3\)

Problem Statement

Calculate the inflection points of \(f(x)=2+3x^2-x^3\) and determine where the function is concave up or concave down.

Solution

PatrickJMT - 1351 video solution

video by PatrickJMT

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\(h(x)=(x^2-1)^3\)

Problem Statement

Calculate the inflection points of \(h(x)=(x^2-1)^3\) and determine where the function is concave up or concave down.

Solution

PatrickJMT - 1352 video 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

MIP4U - 2290 video solution

video by MIP4U

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Practice Instructions

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

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

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