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Unit 5

5.7 Using the Second Derivative Test to Determine Extrema

1 min readjune 8, 2020

Sumi Vora


🎥Watch: AP Calculus AB/BC - Concavity and f, f', f''

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Concavity

@sander-o

Nov 11, 2020

f, f', f''

@jenni513843

Nov 11, 2020

Using the Second Derivative Test

If we combine our knowledge of first derivatives and second derivatives, we find that we can use the second derivative to determine whether a critical point is a relative minimum or relative maximum. ❗️

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(555).png?alt=media&token=267258be-fb61-4885-816b-241e195dcde0

In other words, if we know that there is a local extreme at a certain point and the graph is concave up at that point, it must be a minimum, and, if the graph is concave down at that point, it must be a maximum. 🙋‍♂️

Example Problems

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(557).png?alt=media&token=aab95edc-0f28-4536-aecf-57fb3f325e26

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(559).png?alt=media&token=942c24a2-a078-486d-bcf7-fed0745e0647

We can find the critical points of implicit functions in a similar manner. The critical points occur when dy/dx = 0.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(562).png?alt=media&token=87671976-7757-4cd8-97df-c2864c434bb5

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(563).png?alt=media&token=451d6dc3-522f-408e-aa0b-a0b51cae8f6e

 

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