Difference between revisions of "Math 22 Concavity and the Second-Derivative Test"

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   1. If <math>f''(x)>0</math> for all <math>x</math> in <math>I</math>, then the graph of <math>f</math> is concave upward on <math>I</math>.
 
   1. If <math>f''(x)>0</math> for all <math>x</math> in <math>I</math>, then the graph of <math>f</math> is concave upward on <math>I</math>.
 
   2. If <math>f''(x)<0</math> for all <math>x</math> in <math>I</math>, then the graph of <math>f</math> is concave downward on <math>I</math>.
 
   2. If <math>f''(x)<0</math> for all <math>x</math> in <math>I</math>, then the graph of <math>f</math> is concave downward on <math>I</math>.
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==Guidelines for Applying the Concavity Test==
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  1. Locate the <math>x</math>-values at which <math>f''(x)=0</math> or <math>f''(x)</math> is undefined.
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  2. Use these <math>x</math>-values to determine the test intervals.
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  3. Determine the sign of <math>f'(x)</math> at an arbitrary number in each test intervals
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  4. Apply the concavity test
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'''Exercises:''' Find the second derivative of <math>f</math> and discuss the concavity of its graph.
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'''1)''' <math>f(x)=x^3+2x^2</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|'''Step 1''': <math>f'(x)=3x^2+4x</math>, so <math>f''(x)=6x=0</math>
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|-
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|'''Step 2''': So <math>x=0</math>, so the test intervals are <math>(-\infty,0)</math> and <math>(0,\infty)</math>
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|-
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|'''Step 3''': Choose <math>x=-1</math> for the interval <math>(-\infty,0)</math>, and <math>x=1</math> for the interval <math>(0,\infty)</math>.
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|-
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|Then we have: <math>f''(-1)=-6<0</math> and <math>f''(1)=6>0</math>
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|-
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|'''Step 4''': By the concavity test, <math>f(x)</math> is concave up in <math>(0,\infty)</math> and <math>f(x)</math> is concave down in <math>(-\infty,0)</math>
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|}
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'''2)''' <math>f(x)=x^4-2x^3+10</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|'''Step 1''': <math>f'(x)=4x^3-6x^2</math>, so <math>f''(x)=12x^2-12x=12x(x-1)=0</math>
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|-
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|'''Step 2''': So, <math>x=0</math> and <math>x=1</math>, so the test intervals are <math>(-\infty,0),(0,1)</math> and <math>(1,\infty)</math>
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|-
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|'''Step 3''': Choose <math>x=-1</math> for the interval <math>(-\infty,0)</math>, <math>x=\frac{1}{2}</math> for the interval <math>(0,1)</math> and <math>x=2</math> for the interval <math>(1,\infty)</math>.
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|-
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|Then we have: <math>f''(-1)=24>0</math>, <math>f''(\frac{1}{2})=-3<0</math> and <math>f''(2)=24>0</math>
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|-
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|'''Step 4''': By the concavity test, <math>f(x)</math> is concave up in <math>(0,\infty)\cup (1,\infty)</math> and <math>f(x)</math> is concave down in <math>(0,1)</math>
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|}
  
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Revision as of 07:04, 31 July 2020

Formal Definition of Concavity

 Let  be differentiable on an open interval . The graph of  is
 1. Concave upward on  when  is increasing on the interval.
 2. Concave downward on  when  is decreasing on the interval.

Test for Concavity

 Let  be a function whose second derivative exists on an open interval 
 1. If  for all  in , then the graph of  is concave upward on .
 2. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(x)<0}
 for all  in , then the graph of  is concave downward on .

Guidelines for Applying the Concavity Test

 1. Locate the -values at which  or  is undefined.
 2. Use these -values to determine the test intervals.
 3. Determine the sign of  at an arbitrary number in each test intervals
 4. Apply the concavity test


Exercises: Find the second derivative of and discuss the concavity of its graph.

1)

Solution:  
Step 1: , so
Step 2: So , so the test intervals are and
Step 3: Choose for the interval , and for the interval .
Then we have: and
Step 4: By the concavity test, is concave up in and is concave down in

2)

Solution:  
Step 1: , so
Step 2: So, and , so the test intervals are and
Step 3: Choose for the interval , for the interval and for the interval .
Then we have: , and
Step 4: By the concavity test, is concave up in and is concave down in

Return to Topics Page

This page were made by Tri Phan