Difference between revisions of "Math 22 Extrema and First Derivative Test"

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!Solution:  
 
!Solution:  
 
|-
 
|-
−
|Step 1: <math>f'(x)=2x+2=2(x+1)=0</math>,  
+
|'''Step 1''': <math>f'(x)=2x+2=2(x+1)=0</math>,  
 
|-
 
|-
−
|Step 2: Critical number is <math>x=-1</math>, so the test intervals are <math>(-\infty,-1)</math> and <math>(-1,\infty)</math>
+
|'''Step 2''': Critical number is <math>x=-1</math>, so the test intervals are <math>(-\infty,-1)</math> and <math>(-1,\infty)</math>
 
|-
 
|-
−
|Step 3: Choose <math>x=-2</math> for the interval <math>(-\infty,-1)</math>, and <math>x=0</math> for the interval <math>(-1,\infty)</math>.
+
|'''Step 3''': Choose <math>x=-2</math> for the interval <math>(-\infty,-1)</math>, and <math>x=0</math> for the interval <math>(-1,\infty)</math>.
 
|-
 
|-
 
|Then we have: <math>f'(-2)=-2<0</math> and <math>f'(0)=2>0</math>
 
|Then we have: <math>f'(-2)=-2<0</math> and <math>f'(0)=2>0</math>
 
|-
 
|-
−
|Step 4: <math>f'(x)</math> is negative to the left of <math>x=-1</math> and positive to the right of <math>x=-1</math>, then <math>f(1)=3</math> is a relative minimum
+
|'''Step 4''': By the first derivative test, <math>f'(x)</math> is negative to the left of <math>x=-1</math> and positive to the right of <math>x=-1</math>, then <math>f(1)=3</math> is a relative minimum
 
|-
 
|-
 
|Therefore, Relative minimum: <math>(1,3)</math>
 
|Therefore, Relative minimum: <math>(1,3)</math>
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!Solution: &nbsp;
 
!Solution: &nbsp;
 
|-
 
|-
−
|Step 1: <math>f'(x)=15x^2-20x=5x(3x-4)=0</math>,  
+
|'''Step 1''': <math>f'(x)=15x^2-20x=5x(3x-4)=0</math>,  
 
|-
 
|-
−
|Step 2: Critical number is <math>x=0</math> and <math>x=\frac{4}{3}</math>, so the test intervals are <math>(-\infty,0),(0,\frac{4}{3})</math></math> and <math>(\frac{4}{3},\infty)</math>
+
|'''Step 2''': Critical number is <math>x=0</math> and <math>x=\frac{4}{3}</math>, so the test intervals are <math>(-\infty,0),(0,\frac{4}{3})</math> and <math>(\frac{4}{3},\infty)</math>
 
|-
 
|-
−
|Step 3: Choose <math>x=-2</math> for the interval <math>(-\infty,-1)</math>, and <math>x=0</math> for the interval <math>(-1,\infty)</math>.
+
|'''Step 3''': Choose <math>x=-1</math> for the interval <math>(-\infty,0)</math>, <math>x=1</math> for the interval <math>(0,\frac{4}{3})</math> and <math>x=2</math> for the interval <math>(\frac{4}{3},\infty)</math>.
 
|-
 
|-
−
|Then we have: <math>f'(-2)=-2<0</math> and <math>f'(0)=2>0</math>
+
|Then we have: <math>f'(-1)=35>0</math>, <math>f'(1)=-5<0</math> and <math>f'(2)=20>0</math>
 +
|-
 +
|'''Step 4''': By the first derivative test, <math>f'(x)</math> is positive to the left of <math>x=0</math> and negative to the right of <math>x=0</math>, then <math>f(0)=3</math> is a relative maximum,
 +
|-
 +
| and <math>f'(x)</math> is negative to the left of <math>x=\frac{4}{3}</math> and positive to the right of <math>x=\frac{4}{3}</math>, then <math>f(\frac{4}{3})=\frac{-79}{27}</math> is a relative minimum.
 
|-
 
|-
−
|Step 4: <math>f'(x)</math> is negative to the left of <math>x=-1</math> and positive to the right of <math>x=-1</math>, then <math>f(1)=3</math> is a relative minimum
+
|Therefore, Relative minimum: <math>(\frac{4}{3},\frac{-79}{27})</math> and Relative maximum: <math>(0,3)</math>
 
|}
 
|}
  
 
==Absolute Extrema==
 
==Absolute Extrema==
−
The page is under Construction
+
  Let <math>f</math> be defined on an interval <math>I</math> containing <math>c</math>.
 +
  1. <math>f(c)</math> is an absolute minimum of <math>f</math> on <math>I</math> when <math>f(c)\le f(x)</math> for every <math>x</math> in <math>I</math>
 +
  2. <math>f(c)</math> is an absolute maximum of <math>f</math> on <math>I</math> when <math>f(c)\ge f(x)</math> for every <math>x</math> in <math>I</math>
 +
 
 +
==Extreme Value Theorem==
 +
 
 +
  If <math>f</math> is continuous on a closed interval <math>[a,b]</math>, then <math>f</math> has both a minimum value and a maximum value on <math>[a,b]</math>.
 +
 
 +
==Guidelines for Finding Extrema on a Closed Interval==
 +
  To find the extrema of a continuous function <math>f</math> on a closed interval <math>[a,b]</math>, use the following steps.
 +
  1. Find all critical numbers of <math>f</math>
 +
  2. Evaluate <math>f</math> at each of its critical number
 +
  3. Evaluate <math>f</math> at each end point <math>a</math> and <math>b</math>
 +
  4. The least of these values is the absolute minimum, and the greatest is the maximum.
 +
 
 +
'''Exercises:''' Find all absolute extrema of the function below
 +
 
 +
'''1)''' <math>f(x)=5-2x^2</math> on <math>[-3,1]</math>
 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|'''Step 1''': <math>f'(x)=-4x=0</math>, So, critical number is <math>x=0</math>
 +
|-
 +
|'''Step 2''': <math>f(0)=5</math>
 +
|-
 +
|'''Step 3''': <math>f(-3)=-13</math> and <math>f(1)=3</math>
 +
|-
 +
|'''Step 4''': Absolute Maximum is <math>5</math> at <math>x=0</math>
 +
|-
 +
|and absolute minimum is <math>-13</math> at <math>x=-3</math>
 +
|}
  
 
[[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]]'''

Latest revision as of 09:17, 30 July 2020

Relative Extrema

 Let  be a function defined at .
 1.  is a relative maximum of  when there exists an interval  containing  such that  for all  in .
 2.  is a relative minimum of  when there exists an interval  containing  such that  for all  in .

If has a relative minimum or relative maximum at , then is a critical number of . That is, either or is undefined.

Relative extrema must occur at critical numbers as shown in picture below.

Relative extrema.png

The First-Derivative Test

 Let  be continuous on the interval  in which  is the only critical number, then
 
 On the interval , if  is negative to the left of  and positive to the right of , then  is a relative minimum.
 
 On the interval , if  is positive to the left of  and negative to the right of , then  is a relative maximum.

Guidelines for Finding Relative Extrema

 1. Find the derivative of 
 2. Find all critical numbers, then determine the test intervals
 3. Determine the sign of  at an arbitrary number in each test intervals
 4. Apply the first derivative test


Exercises: Find all relative extrema of the functions below

1)

Solution:  
Step 1: ,
Step 2: Critical number is , so the test intervals are and
Step 3: Choose for the interval , and for the interval .
Then we have: and
Step 4: By the first derivative test, is negative to the left of and positive to the right of , then 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(1)=3} is a relative minimum
Therefore, Relative minimum:
(Note: 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)} in this case is a parabola so our answer makes sense)

2)

Solution:  
Step 1: ,
Step 2: Critical number is 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 x=0} and , so the test intervals are 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 (-\infty,0),(0,\frac{4}{3})} and
Step 3: Choose 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 x=-1} for the interval 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 (-\infty,0)} , 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 x=1} for the interval 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 (0,\frac{4}{3})} and 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 x=2} for the interval 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 (\frac{4}{3},\infty)} .
Then we have: 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'(-1)=35>0} , 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'(1)=-5<0} and 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'(2)=20>0}
Step 4: By the first derivative test, 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)} is positive to the left of 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 x=0} and negative to the right of 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 x=0} , then 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(0)=3} is a relative maximum,
and 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)} is negative to the left of 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 x=\frac{4}{3}} and positive to the right of 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 x=\frac{4}{3}} , then 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(\frac{4}{3})=\frac{-79}{27}} is a relative minimum.
Therefore, Relative minimum: 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 (\frac{4}{3},\frac{-79}{27})} and Relative maximum: 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 (0,3)}

Absolute Extrema

 Let 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}
 be defined on an interval 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 I}
 containing 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 c}
.
 1. 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(c)}
 is an absolute minimum of 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}
 on 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 I}
 when 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(c)\le f(x)}
 for every 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 x}
 in 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 I}

 2. 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(c)}
 is an absolute maximum of 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}
 on 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 I}
 when 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(c)\ge f(x)}
 for every 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 x}
 in 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 I}

Extreme Value Theorem

 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}
 is continuous on a closed interval 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 [a,b]}
, then 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}
 has both a minimum value and a maximum value on 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 [a,b]}
.

Guidelines for Finding Extrema on a Closed Interval

 To find the extrema of a continuous function 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}
 on a closed interval 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 [a,b]}
, use the following steps.
 1. Find all critical numbers of 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}

 2. Evaluate 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}
 at each of its critical number
 3. Evaluate 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}
 at each end point 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 a}
 and 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 b}

 4. The least of these values is the absolute minimum, and the greatest is the maximum.

Exercises: Find all absolute extrema of the function below

1) 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)=5-2x^2} on 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 [-3,1]}

Solution:  
Step 1: 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)=-4x=0} , So, critical number is 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 x=0}
Step 2: 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(0)=5}
Step 3: 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(-3)=-13} and 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(1)=3}
Step 4: Absolute Maximum is 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 5} at 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 x=0}
and absolute minimum is 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 -13} at 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 x=-3}

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This page were made by Tri Phan

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