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Unit 5

5.7 Using the Second Derivative Test to Determine Extrema

1 min readโ€ขjune 8, 2020

sumi-vora

Sumi Vora


AP Calculus AB/BCย โ™พ๏ธ

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๐ŸŽฅWatch: AP Calculus AB/BC - Concavity and f, f', f''

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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.ย โ—๏ธ
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. ๐Ÿ™‹โ€โ™‚๏ธ

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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