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

Revision as of 10:05, 30 July 2020

Relative Extrema

 Let  $f$  be a function defined at  $c$ .
 1.  $f(c)$  is a relative maximum of  $f$  when there exists an interval  $(a,b)$  containing  $c$  such that  $f(x)\leq f(c)$  for all  $x$  in  $(a,b)$ .
 2.  $f(c)$  is a relative minimum of  $f$  when there exists an interval  $(a,b)$  containing  $c$  such that  $f(x)\geq f(c)$  for all  $x$  in  $(a,b)$ .

If $f$ has a relative minimum or relative maximum at $x=c$ , then $c$ is a critical number of $f$ . That is, either Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(c)=0} or $f'(c)$ is undefined.

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

The First-Derivative Test

 Let  $f$  be continuous on the interval  $(a,b)$  in which  $c$  is the only critical number, then
 
 On the interval  $(a,b)$ , if  $f'(x)$  is negative to the left of  $x=c$  and positive to the right of  $x=c$ , then  $f(c)$  is a relative minimum.
 
 On the interval  $(a,b)$ , if  $f'(x)$  is positive to the left of  $x=c$  and negative to the right of  $x=c$ , then  $f(c)$  is a relative maximum.

Guidelines for Finding Relative Extrema

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

Exercises: Find all relative extrema of the functions below

1) $f(x)=x^{2}+2x$

Solution:
Step 1: $f'(x)=2x+2=2(x+1)=0$ ,
Step 2: Critical number is $x=-1$ , so the test intervals are $(-\infty ,-1)$ and $(-1,\infty )$
Step 3: Choose $x=-2$ for the interval $(-\infty ,-1)$ , and $x=0$ for the interval $(-1,\infty )$ .
Then we have: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(-2)=-2<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'(0)=2>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 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=-1} 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=-1} , 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: 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 (1,3)}
(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) 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)=5x^3-10x^2+3}

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)=15x^2-20x=5x(3x-4)=0} ,
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 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}} , 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 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)}
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

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@@ Line 39: / Line 39: @@
 |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>
@@ Line 56: / Line 56: @@
 |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: <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 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.
+|-
+|Therefore, Relative minimum: <math>(\frac{4}{3},\frac{-79}{27})</math> and Relative maximum: <math>(0,3)</math>
 |}

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

Revision as of 10:05, 30 July 2020

Contents

Relative Extrema

The First-Derivative Test

Guidelines for Finding Relative Extrema

Absolute Extrema

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