๐Ÿ“š

All Subjects

ย >ย 

โ™พ๏ธย 

AP Calc

ย >ย 

โœจ

Unit 5

5.7 Using the Second Derivative Test to Determine Extrema

1 min readโ€ขjune 8, 2020

Sumi Vora


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

Resources:

Concavity

@sander-o

โ€ขNov 11, 2020

f, f', f''

@jenni513843

โ€ขNov 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.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

ย 

Was this guide helpful?

Join us on Discord

Thousands of students are studying with us for the AP Calculus AB/BC exam.

join now

Browse Study Guides By Unit

โœ๏ธ
Free Response Questions (FRQ)

๐Ÿง
Multiple Choice Questions (MCQ)

โ™พ
Unit 10: Infinite Sequences and Series (BC Only)

๐Ÿ‘‘
Unit 1: Limits & Continuity

๐Ÿค“
Unit 2: Differentiation: Definition & Fundamental Properties

๐Ÿค™๐Ÿฝ
Unit 3: Differentiation: Composite, Implicit & Inverse Functions

๐Ÿ‘€
Unit 4: Contextual Applications of the Differentiation

๐Ÿ”ฅ
Unit 6: Integration and Accumulation of Change

๐Ÿ’Ž
Unit 7: Differential Equations

๐Ÿถ
Unit 8: Applications of Integration

๐Ÿฆ–
Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)