Difference between revisions of "009A Sample Final 3, Problem 6"

Revision as of 20:37, 6 March 2017

Let

f(x)=4+8x^{3}-x^{4}

(a) Over what $x$ -intervals is $f$ increasing/decreasing?

(b) Find all critical points of $f$ and test each for local maximum and local minimum.

(c) Over what $x$ -intervals is $f$ concave up/down?

(d) Sketch the shape of the graph of $f.$

Foundations:
1. $f(x)$ is increasing when $f'(x)>0$ and $f(x)$ is decreasing when $f'(x)<0.$
2. The First Derivative Test tells us when we have a local maximum or local minimum.
3. $f(x)$ is concave up when $f''(x)>0$ and $f(x)$ is concave down when $f''(x)<0.$

Solution:

(a)

Step 1:
We start by taking the derivative of $f(x).$ We have $f'(x)=24x^{2}-4x^{3}.$
Now, we set $f'(x)=0.$ So, we have $0=4x^{2}(6-x).$
Hence, we have $x=0$ and $x=6.$
So, these values of $x$ break up the number line into 3 intervals: $(-\infty ,0),(0,6),(6,\infty ).$

Step 2:
To check whether the function is increasing or decreasing in these intervals, we use testpoints.
For $x=-1,~f'(x)=28>0.$
For $x=1,~f'(x)=20>0.$
For $x=7,~f'(x)=-196<0.$
Thus, $f(x)$ is increasing on $(-\infty ,6)$ and decreasing on $(6,\infty ).$

(b)

Step 1:
The critical points of $f(x)$ occur at $x=0$ 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=6.}
Plugging these values into 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),} we get the critical points
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 (0,4)} 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 (6,436).}

Step 2:
Using the first derivative test and the information from part (a),
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 (0,4)} is not a local minimum or local 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 (6,436)} is a local maximum.

(c)

Step 1:
To find the intervals when the function is concave up or concave down, we need to find 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).}
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''(x)=48x-12x^2.}
We set 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.}
So, 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 0=12x(4-x).} Hence, 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=4} .
This value breaks up the number line into three intervals: 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),(0,4),(4,\infty).}

Step 2:
Again, we use test points in these three intervals.
For 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,} 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''(x)=-60<0.}
For 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,} 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''(x)=48>0.}
For 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=5,} 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''(x)=-60<0.}
Thus, 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 concave up on 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 (0,4)} and concave down on 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)\cup (4,\infty).}

(d):
Insert graph

Final Answer:
(a) 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 increasing 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 (-\infty,6)} and decreasing 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 (6,\infty).}
(b) The critical points 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 (0,4)} 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 (6,436).}
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 (0,4)} is not a local minimum or local 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 (6,436)} is a local maximum.
(c) 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 concave up on 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 (0,4)} and concave down on 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)\cup (4,\infty).}
(d) See above

Return to Sample Exam

@@ Line 81: / Line 81: @@
 !Step 1: &nbsp;
 |-
-|
+|To find the intervals when the function is concave up or concave down, we need to find &nbsp;<math style="vertical-align: -5px">f''(x).</math>
 |-
-|
+|We have &nbsp;<math style="vertical-align: -5px">f''(x)=48x-12x^2.</math>
 |-
-|
+|We set &nbsp;<math style="vertical-align: -5px">f''(x)=0.</math>
 |-
-|
+|So, we have &nbsp;<math style="vertical-align: -1px">0=12x(4-x).</math>&nbsp; Hence, &nbsp;<math style="vertical-align: 0px">x=0</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">x=4</math>.
 |-
-|
+|This value breaks up the number line into three intervals: &nbsp;<math style="vertical-align: -5px">(-\infty,0),(0,4),(4,\infty).</math>
 |}
@@ Line 95: / Line 95: @@
 !Step 2: &nbsp;
 |-
-|
+|Again, we use test points in these three intervals.
 |-
-|
+|For &nbsp;<math style="vertical-align: -5px">x=-1,</math>&nbsp; we have &nbsp;<math style="vertical-align: -5px">f''(x)=-60<0.</math>
 |-
-|
+|For &nbsp;<math style="vertical-align: -5px">x=1,</math>&nbsp; we have &nbsp;<math style="vertical-align: -5px">f''(x)=48>0.</math>
 |-
-|
+|For &nbsp;<math style="vertical-align: -5px">x=5,</math>&nbsp; we have &nbsp;<math style="vertical-align: -5px">f''(x)=-60<0.</math>
+|-
+|Thus, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is concave up on the interval &nbsp;<math style="vertical-align: -5px">(0,4)</math>&nbsp; and concave down on the interval &nbsp;<math style="vertical-align: -5px">(-\infty,0)\cup (4,\infty).</math>
 |}
@@ Line 120: / Line 122: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -4px">(0,4)</math>&nbsp; is not a local minimum or local maximum and &nbsp;<math style="vertical-align: -5px">(6,436)</math>&nbsp; is a local maximum.
 |-
-|&nbsp; &nbsp;'''(c)'''&nbsp; &nbsp;
+|&nbsp; &nbsp;'''(c)'''&nbsp; &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is concave up on the interval &nbsp;<math style="vertical-align: -5px">(0,4)</math>&nbsp; and concave down on the interval &nbsp;<math style="vertical-align: -5px">(-\infty,0)\cup (4,\infty).</math>
 |-
 |&nbsp; &nbsp;'''(d)'''&nbsp; &nbsp; See above
 |}
 [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Final 3, Problem 6"

Revision as of 20:37, 6 March 2017

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