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17Calculus Derivatives - The Second Derivative Test

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Second Derivative Test

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.

Table Format For The Second Derivative Test

Critical Point

\( x = c_1 \)

\( x = c_2 \)

\( x = c_3 \)

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 [1min-30secs]

video by PatrickJMT

Okay, time for some practice problems.

Practice

Unless otherwise instructed, use the second derivative test to determine maximums and minimums of these functions.

\(f(x)=x^3-3x+1\)

Problem Statement

Use the second derivative test to determine the extrema for \(f(x)=x^3-3x+1\). If the second derivative test is inconclusive, use the first derivative test.

Solution

Krista King Math - 1348 video solution

video by Krista King Math

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Suppose \(f(x)\) has a critical point at \(x=4\) and \(f''(4) = -3\). What can be said about \(f(x)\) at \(x=4\)?

Problem Statement

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

PatrickJMT - 1354 video solution

video by PatrickJMT

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\(\displaystyle{f(x)=\frac{x}{x^2+4}}\)

Problem Statement

Use the second derivative test to determine the extrema for \(\displaystyle{f(x)=\frac{x}{x^2+4}}\). If the second derivative test is inconclusive, use the first derivative test.

Solution

PatrickJMT - 1353 video solution

video by PatrickJMT

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Practice Search

Practice Instructions

Unless otherwise instructed, use the second derivative test to determine maximums and minimums of 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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