009A Sample Final A, Problem 10 - Revision history

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MathAdmin: Created page with "10. Consider the function $f(x)=2x^{3}+4x+\sqrt{2}.$
(a)..."

2015-03-24T04:13:27Z

Created page with "10. Consider the function <math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math> (a)..."

New page

10. Consider the function
 
<math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math>
 
   (a) Use the Intermediate Value Theorem to show that <math style="vertical-align: -14%;">f(x)</math> has at
least one zero.
 
   (b) Use Rolle's Theorem to show that <math style="vertical-align: -14%;">f(x)</math> has exactly one zero. 

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations  
|''The Intermediate Value Theorem.'' If ''f''(''x'') is a continuous function on the interval [''a'',''b''], and if ''f''(''a'') < ''f''(''b''), then for any ''y'' such that
|-
|<math>f(a)\leq y\leq f(b),</math>
|-
|there exists a <math>c\in [a,b]</math> such that <math>f(c)=y</math>.
|}

@@ Line 21: / Line 21: @@
 |}
-'''Solution:'''
+&nbsp;'''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

← Older revision		Revision as of 05:24, 24 March 2015
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	\|}		\|}

−	[[009A_Sample_Final_A\|'''Return to Sample Exam''']]	+	[[009A_Sample_Final_A\|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 05:23, 24 March 2015
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	\|However, we know that ''f'' '(''x'') =6''x''<sup>2</sup> + 4, which is greater than zero for all ''x''. This contradicts that ''f'''(''z'') = 0, so the assumption that another root exists must be incorrect.		\|However, we know that ''f'' '(''x'') =6''x''<sup>2</sup> + 4, which is greater than zero for all ''x''. This contradicts that ''f'''(''z'') = 0, so the assumption that another root exists must be incorrect.
	\|}		\|}
		+
		+	[[009A_Sample_Final_A\|'''Return to Sample Exam''']]

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 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Foundations &nbsp;
+! Foundations: &nbsp;
 |-
 |<u>'''The Intermediate Value Theorem</u>.''' ''If f is a continuous function on the interval ''[''a,b'']'', and if  f''(''a'')'' &le; f''(''b'')'', then for any y such that f''(''a'')'' &le; y &le; f''(''b'')'', then there exists a c &isin; ''[''a,b'']'' such that f''(''c'')'' = y. Similarly, if f''(''a'')'' &ge; f''(''b'')'', then for any y such that f''(''a'')'' &ge; y &ge; f''(''b'')'', then there exists a c &isin; ''[''a,b'']'' such that f''(''c'')'' = y.''

@@ Line 11: / Line 11: @@
 ! Foundations &nbsp;
 |-
-|<u>'''The Intermediate Value Theorem</u>.''' ''If f''(''x'')'' is a continuous function on the interval ''[''a,b'']'', and if  f''(''a'')'' &le; f''(''b'')'', then for any y such that f''(''a'')'' &le; y &le; f''(''b'')'', then there exists a c &isin; ''[''a,b'']'' such that f''(''c'')'' = y.''
+|<u>'''The Intermediate Value Theorem</u>.''' ''If f''(''x'')'' is a continuous function on the interval ''[''a,b'']'', and if  f''(''a'')'' &le; f''(''b'')'', then for any y such that f''(''a'')'' &le; y &le; f''(''b'')'', then there exists a c &isin; ''[''a,b'']'' such that f''(''c'')'' = y. Similarly, if f''(''a'')'' &ge; f''(''b'')'', then for any y such that f''(''a'')'' &ge; y &ge; f''(''b'')'', then there exists a c &isin; ''[''a,b'']'' such that f''(''c'')'' = y.''
 |}