MAT 3749 Introduction to Analysis
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Transcript of MAT 3749 Introduction to Analysis
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MAT 3749Introduction to Analysis
Section 2.3 Part III
The Mean Value Theorem
http://myhome.spu.edu/lauw
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Important Result
)()( xgxf
Cxgxf )()(
a b
)(xfy
)(xgy
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Preview
Extreme Value Theorem Fermat’s Theorem Rolle’s Theorem The Mean Value Theorem
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References
Section 2.3
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Maximum Value
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Local Maximum
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T or F
An absolute max is a local max.
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The Extreme Value Theorem
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Fermat’s Theorem
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Lemma (HW)
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Fermat’s Theorem
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Conceptual Diagrams
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Fermat’s Theorem
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Fermat’s Theorem
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Fermat’s Theorem
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Fermat’s Theorem
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Proof
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Proof
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Proof
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Proof
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Rolle’s Theorem
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Proof
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Proof
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The Mean Value Theorem
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Proof
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The Mean Value Theorem
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The Mean Value Theorem
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The Mean Value Theorem
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Theorem (Consequence)
If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b).
Q: Can we apply the MVT directly?
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Corollary (Important)
)()( xgxf Cxgxf )()(
a b
)(xfy
)(xgy
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Corollary (Important)
)()( xgxf Cxgxf )()(
a b
)(xfy
)(xgy C