MathAdmin at 00:34, 18 May 2015

2015-05-18T00:34:03Z

MathAdmin at 19:42, 17 May 2015

2015-05-17T19:42:46Z

MathAdmin: Created page with " Find all relative extrema and points of inflection for the function $g(x) = x^3 - 3x$ . Be sure to give coordinate p..."

2015-05-17T19:42:07Z

Created page with "<span class="exam"> Find all relative extrema and points of inflection for the function <math style="vertical-align: -16%">g(x) = x^3 - 3x</math>. Be sure to give coordinate p..."

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<span class="exam">
Find all relative extrema and points of inflection for the function <math style="vertical-align: -16%">g(x) = x^3 - 3x</math>. Be sure to give coordinate pairs for each point. You do not need to draw the graph. Explain how you know which point is the local minimum and which is the local maximum (i.e., which test did you use?).

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|Since our function is a polynomial, the relative extrema occur when the first derivative is zero. We then have two choices for finding if it is a local maximum or minimum:
|-
|'''Second Derivative Test:''' If the first derivative at a point <math style="vertical-align: -12%">x_0</math> is <math style="vertical-align: 0%">0</math>, and the second derivative is negative (indicating it is concave-down, like an upside-down parabola), then the point <math style="vertical-align: -20%">\left(x_0,f(x_0)\right)</math> is a local maximum.
|-
|On the other hand, if the second derivative is positive, the point <math style="vertical-align: -20%">\left(x_0,f(x_0)\right)</math> is a local minimum. You can also use the first derivative test, but it is usually a bit more work! For '''inflection points''', we need to find when the second derivative is zero, as well as check that the second derivative "splits" on both sides.
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|'''Find the first and second derivatives:''' Based on our function, we have
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|
::<math>g\,'(x)\,=\,3x^2-3.</math>
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|Similarly, from the first derivative we find
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|
::<math>g\,''(x)\,=\,6x.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|'''Find the roots of the derivatives:''' We can rewrite the first derivative as
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|
::<math>g\,'(x)\,=\,3x^2-3\,=\,3(x^2-1)\,=\,3(x-1)(x+1),</math>
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|from which it should be clear we have roots <math style="vertical-align: 0%">\pm1</math>.
|-
|On the other hand, for the second derivative, we have only a single root: <math style="vertical-align: 0%">x=0</math>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
|'''Test the potential extrema:''' We know that <math style="vertical-align: 0%">x=\pm1</math> are the candidates. We check the second derivative, finding
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::<math>g\,''(1)\,=\,6\,>\,0,</math>
|-
|while
|-
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::<math>g\,''(-1)\,=\,-6\,<\,0.</math>
|-
|Note that
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::<math>g(1)\,=\,1-3\,=\,-2,</math>
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|while
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::<math>g(-1)\,=\,-1+3\,=\,2.</math>
|-
|By the second derivative test, the point <math style="vertical-align: -20%">(1,g(1))=\left(1,-2\right)</math> is a relative minimum, while the point <math style="vertical-align: -20%">(-1,g(-1))=(-1,2)</math> is a relative maximum.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 4:  
|-
|'''Test the potential inflection point:''' We know that <math style="vertical-align: -25%">g\,''(0)=0</math>. On the other hand, it should be clear that if <math style="vertical-align: 0%">x<0</math>, then <math style="vertical-align: -20%">g\,''(x)<0</math>. Similarly, if <math style="vertical-align: -20%">x>0,</math> then <math style="vertical-align: -20%">g\,''(x)>0</math>. Thus, the second derivative "splits" around <math style="vertical-align: 0%">x=0</math>  (i.e., changes sign), so the point <math style="vertical-align: -25%">\left(0,g(0)\right)=(0,0)</math>  is an inflection point.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|There is a local minimum at <math style="vertical-align: -20%">(1,-2)</math>, a local maximum at <math style="vertical-align: -20%">(-1,2)</math> and an inflection point at <math style="vertical-align: -20%">(0,0).</math>
|}

[[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

@@ Line 44: / Line 44: @@
 !Step 3: &nbsp;
 |-
-|'''Test the potential extrema:''' We know that <math style="vertical-align: 0%">x=\pm1</math> are the candidates.  We check the second derivative, finding
+|'''Test the potential extrema:''' We know that <math style="vertical-align: -2%">x=\pm1</math> are the candidates.  We check the second derivative, finding
 |-
+|

@@ Line 70: / Line 70: @@
 !Step 4: &nbsp;
 |-
-|'''Test the potential inflection point:''' We know that <math style="vertical-align: -25%">g\,''(0)=0</math>. On the other hand, it should be clear that if <math style="vertical-align: 0%">x<0</math>, then <math style="vertical-align: -20%">g\,''(x)<0</math>.  Similarly, if <math style="vertical-align: -20%">x>0,</math> then <math style="vertical-align: -20%">g\,''(x)>0</math>. Thus, the second derivative "splits" around <math style="vertical-align: 0%">x=0</math>&thinsp; (i.e., changes sign), so the point <math style="vertical-align: -25%">\left(0,g(0)\right)=(0,0)</math>&thinsp; is an inflection point.
+|'''Test the potential inflection point:''' We know that <math style="vertical-align: -25%">g\,''(0)=0</math>. On the other hand, it should be clear that if <math style="vertical-align: 0%">x<0</math>, then <math style="vertical-align: -20%">g\,''(x)<0</math>.  Similarly, if <math style="vertical-align: -20%">x>0,</math> then <math style="vertical-align: -20%">g\,''(x)>0</math>. Thus, the second derivative "splits" around <math style="vertical-align: -3%">x=0</math>&thinsp; (i.e., changes sign), so the point <math style="vertical-align: -25%">\left(0,g(0)\right)=(0,0)</math>&thinsp; is an inflection point.
 |}

022 Exam 2 Sample B, Problem 9 - Revision history

MathAdmin at 00:34, 18 May 2015

MathAdmin at 19:42, 17 May 2015

MathAdmin: Created page with " Find all relative extrema and points of inflection for the function g(x) = x^3 - 3x. Be sure to give coordinate p..."

MathAdmin: Created page with " Find all relative extrema and points of inflection for the function $g(x) = x^3 - 3x$ . Be sure to give coordinate p..."