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

Revision as of 13:42, 18 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 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=4x^2(6-x).}
Hence, 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 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.}
So, these values 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} break up the number line into 3 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),(0,6),(6,\infty).}

Step 2:
To check whether the function is increasing or decreasing in these intervals, we use testpoints.
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,~f'(x)=28>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,~f'(x)=20>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=7,~f'(x)=-196<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 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)

Step 1:
The critical points 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(x)} occur 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 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, $f(x)$ is concave up on the interval $(0,4)$ and concave down on the interval $(-\infty ,0)\cup (4,\infty ).$

(d):
Insert graph

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

Return to Sample Exam

@@ Line 29: / Line 29: @@
 !Step 1: &nbsp;
 |-
-|We start by taking the derivative of &nbsp;<math style="vertical-align: -5px">f(x).</math>&nbsp; We have &nbsp;<math style="vertical-align: -5px">f'(x)=24x^2-4x^3.</math>
+|We start by taking the derivative of &nbsp;<math style="vertical-align: -5px">f(x).</math>&nbsp;
 |-
-|Now, we set &nbsp;<math style="vertical-align: -5px">f'(x)=0.</math>&nbsp; So, we have &nbsp;<math style="vertical-align: -6px">0=4x^2(6-x).</math>
+|We have &nbsp;<math style="vertical-align: -5px">f'(x)=24x^2-4x^3.</math>
+|-
+|Now, we set &nbsp;<math style="vertical-align: -5px">f'(x)=0.</math>&nbsp; So, we have
+|-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -6px">0=4x^2(6-x).</math>
 |-
 |Hence, we have &nbsp;<math style="vertical-align: 0px">x=0</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">x=6.</math>
 |-
-|So, these values of &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; break up the number line into 3 intervals: &nbsp;<math style="vertical-align: -5px">(-\infty,0),(0,6),(6,\infty).</math>
+|So, these values of &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; break up the number line into 3 intervals:
+|-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">(-\infty,0),(0,6),(6,\infty).</math>
 |}
@@ Line 87: / Line 93: @@
 |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>.
+|So, we have
+|-
+|&nbsp; &nbsp; &nbsp; &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:
 |-
-|This value breaks up the number line into three intervals: &nbsp;<math style="vertical-align: -5px">(-\infty,0),(0,4),(4,\infty).</math>
+|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">(-\infty,0),(0,4),(4,\infty).</math>
 |}

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

Revision as of 13:42, 18 March 2017

Navigation menu

Search