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

Revision as of 19:50, 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 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.}

Solution:

(a)

Step 2:

(b)

Step 1:

Step 2:

(c)

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