Math 22 Concavity and the Second-Derivative Test - Revision history

Tphan046: /* The Second-Derivative Test */

2020-07-31T15:29:43Z

The Second-Derivative Test

2020-07-31T15:28:48Z

2020-07-31T15:23:20Z

2020-07-31T15:20:35Z

Points of Inflection

2020-07-31T15:20:15Z

2020-07-31T15:07:59Z

Points of Inflection

2020-07-31T15:07:48Z

Points of Inflection

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2020-07-31T15:04:42Z

Guidelines for Applying the Concavity Test

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 |By the second derivative test, <math>f(0)=-5</math> is relative maximum and <math>f(1)=0</math> is relative minimum
 |}

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 . If <math>f''(c)=0</math>, then the test fails. Use the first derivative test.
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 Therefore, <math>(0,10)</math> and <math>(1,9)</math> are points of inflection.
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    then either <math>f''(c)=0</math> or <math>f''(c)</math> is undefined.
-In exercises 1, at <math>x=0</math>, the concavity changes from concave down to concave up, so <math>(0,f(0))</math> is a point of inflection.
+In '''exercises 1''', at <math>x=0</math>, the concavity changes from concave down to concave up, so <math>(0,f(0))</math> is a point of inflection.
 Therefore, <math>(0,0)</math> is a point of inflection
 In exercises 2, at <math>x=0</math> and <math>x=1</math>, the concavity changes from concave up to concave down and from concave down to concave up, respectively. So, <math>(0,f(0))</math> and <math>(1,f(1))</math> are points of inflection.
 Therefore, <math>(0,10)</math> and <math>(1,9)</math> are points of inflection.

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    then either <math>f''(c)=0</math> or <math>f''(c)</math> is undefined.
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    then the point is a point of inflection.
-   If <math>(c,f(c))</math> is a point of inflection of the graph of <math>f</math>, then either <math>f''(c)=0</math> or <math>f''(c)</math> is undefined.
+   If <math>(c,f(c))</math> is a point of inflection of the graph of <math>f</math>,
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 |}
 ==Points of Inflection==
-   If the graph of a continuous function has a tangent line at a point where its concavity changes from upward to downward (or downward to upward), then the point is a point of inflection.
+   If the graph of a continuous function has a tangent line at a point
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 |Then we have: <math>f''(-1)=24>0</math>, <math>f''(\frac{1}{2})=-3<0</math> and <math>f''(2)=24>0</math>
 |-
-|'''Step 4''': By the concavity test, <math>f(x)</math> is concave up in <math>(0,\infty)\cup (1,\infty)</math> and <math>f(x)</math> is concave down in <math>(0,1)</math>
+|'''Step 4''': By the concavity test, <math>f(x)</math> is concave up in <math>(-\infty,0)\cup (1,\infty)</math> and <math>f(x)</math> is concave down in <math>(0,1)</math>
 |}

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 . If <math>f''(x)>0</math> for all <math>x</math> in <math>I</math>, then the graph of <math>f</math> is concave upward on <math>I</math>.
 . If <math>f''(x)<0</math> for all <math>x</math> in <math>I</math>, then the graph of <math>f</math> is concave downward on <math>I</math>.
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 '''This page were made by [[Contributors|Tri Phan]]'''