Difference between revisions of "009A Sample Final A, Problem 10"
(Created page with "<span style="font-size:135%"><font face=Times Roman>10. Consider the function <math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math> <br> (a)...") |
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| − | |< | + | |<u>'''The Intermediate Value Theorem</u>.''' ''If f''(''x'')'' is a continuous function on the interval ''[''a,b'']'', and if f''(''a'')'' ≤ f''(''b'')'', then for any y such that f''(''a'')'' ≤ y ≤ f''(''b'')'', then there exists a c ∈ ''[''a,b'']'' such that f''(''c'')'' = y.'' |
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Revision as of 20:26, 23 March 2015
10. Consider the function
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)=2x^{3}+4x+\sqrt{2}.}
(a) Use the Intermediate Value Theorem to show that 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)}
has at
least one zero.
(b) Use Rolle's Theorem to show that 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)}
has exactly one zero.
| Foundations |
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| 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 f(a) ≤ y ≤ f(b), then there exists a c ∈ [a,b] such that f(c) = y. |