MathAdmin at 21:04, 23 May 2017

2017-05-23T21:04:34Z

MathAdmin: Created page with " Let :: $f(x)=4+8x^3-x^4$ (a) Over what $x$ -intervals is
2017-04-10T16:30:56Z

Created page with " Let ::<math>f(x)=4+8x^3-x^4</math> (a) Over what <math style="vertical-align: 0px">x</math>-intervals is <math style="vert..."

New page
 Let

::<math>f(x)=4+8x^3-x^4</math>

(a) Over what  <math style="vertical-align: 0px">x</math>-intervals is  <math style="vertical-align: -4px">f</math>  increasing/decreasing?

(b) Find all critical points of  <math style="vertical-align: -4px">f</math>  and test each for local maximum and local minimum.

(c) Over what  <math style="vertical-align: 0px">x</math>-intervals is  <math style="vertical-align: -4px">f</math>  concave up/down?

(d) Sketch the shape of the graph of  <math style="vertical-align: -4px">f.</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''1.''' <math style="vertical-align: -5px">f(x)</math>  is increasing when  <math style="vertical-align: -5px">f'(x)>0</math>  and  <math style="vertical-align: -5px">f(x)</math>  is decreasing when  <math style="vertical-align: -5px">f'(x)<0.</math>
|-
|'''2. The First Derivative Test''' tells us when we have a local maximum or local minimum.
|-
|'''3.''' <math style="vertical-align: -5px">f(x)</math>  is concave up when  <math style="vertical-align: -5px">f''(x)>0</math>  and  <math style="vertical-align: -5px">f(x)</math>  is concave down when  <math style="vertical-align: -5px">f''(x)<0.</math>
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|We start by taking the derivative of  <math style="vertical-align: -5px">f(x).</math> 
|-
|We have  <math style="vertical-align: -5px">f'(x)=24x^2-4x^3.</math>
|-
|Now, we set  <math style="vertical-align: -5px">f'(x)=0.</math>  So, we have
|-
|       <math style="vertical-align: -6px">0=4x^2(6-x).</math>
|-
|Hence, we have  <math style="vertical-align: 0px">x=0</math>  and  <math style="vertical-align: -1px">x=6.</math>
|-
|So, these values of  <math style="vertical-align: 0px">x</math>  break up the number line into 3 intervals:
|-
|       <math style="vertical-align: -5px">(-\infty,0),(0,6),(6,\infty).</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|To check whether the function is increasing or decreasing in these intervals, we use testpoints.
|-
|For  <math style="vertical-align: -5px">x=-1,~f'(x)=28>0.</math>
|-
|For  <math style="vertical-align: -5px">x=1,~f'(x)=20>0.</math>
|-
|For  <math style="vertical-align: -5px">x=7,~f'(x)=-196<0.</math>
|-
|Thus,  <math style="vertical-align: -5px">f(x)</math>  is increasing on  <math style="vertical-align: -5px">(-\infty,6)</math>  and decreasing on  <math style="vertical-align: -5px">(6,\infty).</math>
|-
|
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|The critical points of  <math style="vertical-align: -5px">f(x)</math>  occur at  <math style="vertical-align: 0px">x=0</math>  and  <math style="vertical-align: 0px">x=6.</math>
|-
|Plugging these values into  <math style="vertical-align: -5px">f(x),</math>  we get the critical points
|-
|       <math style="vertical-align: -4px">(0,4)</math>  and  <math style="vertical-align: -4px">(6,436).</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Using the first derivative test and the information from part (a),
|-
| <math style="vertical-align: -4px">(0,4)</math>  is not a local minimum or local maximum and
|-
| <math style="vertical-align: -4px">(6,436)</math>  is a local maximum.
|}

'''(c)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|To find the intervals when the function is concave up or concave down, we need to find  <math style="vertical-align: -5px">f''(x).</math>
|-
|We have  <math style="vertical-align: -5px">f''(x)=48x-12x^2.</math>
|-
|We set  <math style="vertical-align: -5px">f''(x)=0.</math>
|-
|So, we have
|-
|       <math style="vertical-align: -1px">0=12x(4-x).</math> 
|-
|Hence,  <math style="vertical-align: 0px">x=0</math>  and  <math style="vertical-align: 0px">x=4</math>.
|-
|This value breaks up the number line into three intervals:
|-
|       <math style="vertical-align: -5px">(-\infty,0),(0,4),(4,\infty).</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Again, we use test points in these three intervals.
|-
|For  <math style="vertical-align: -5px">x=-1,</math>  we have  <math style="vertical-align: -5px">f''(x)=-60<0.</math>
|-
|For  <math style="vertical-align: -5px">x=1,</math>  we have  <math style="vertical-align: -5px">f''(x)=48>0.</math>
|-
|For  <math style="vertical-align: -5px">x=5,</math>  we have  <math style="vertical-align: -5px">f''(x)=-60<0.</math>
|-
|Thus,  <math style="vertical-align: -5px">f(x)</math>  is concave up on the interval  <math style="vertical-align: -5px">(0,4)</math>  and concave down on the interval  <math style="vertical-align: -5px">(-\infty,0)\cup (4,\infty).</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(d):  
|-
|Insert graph
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|   '''(a)'''    <math style="vertical-align: -5px">f(x)</math>  is increasing on  <math style="vertical-align: -5px">(-\infty,6)</math>  and decreasing on  <math style="vertical-align: -5px">(6,\infty).</math>
|-
|   '''(b)'''    The critical points are <math style="vertical-align: -4px">(0,4)</math>  and  <math style="vertical-align: -4px">(6,436).</math>
|-
|           <math style="vertical-align: -4px">(0,4)</math>  is not a local minimum or local maximum and  <math style="vertical-align: -5px">(6,436)</math>  is a local maximum.
|-
|   '''(c)'''   <math style="vertical-align: -5px">f(x)</math>  is concave up on the interval  <math style="vertical-align: -5px">(0,4)</math>  and concave down on the interval  <math style="vertical-align: -5px">(-\infty,0)\cup (4,\infty).</math>
|-
|   '''(d)'''    See above
|}
[[009A_Sample_Final_3|'''Return to Sample Exam''']]

@@ Line 121: / Line 121: @@
 !(d): &nbsp;
 |-
-|Insert graph
+|[[File:009A_SF3_6.jpg |center|400px]]
 |}