MAT 3749 Introduction to Analysis
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Transcript of MAT 3749 Introduction to Analysis
MAT 3749Introduction to Analysis
Section 2.3 Part III
The Mean Value Theorem
http://myhome.spu.edu/lauw
Important Result
)()( xgxf
Cxgxf )()(
a b
)(xfy
)(xgy
Preview
Extreme Value Theorem Fermat’s Theorem Rolle’s Theorem The Mean Value Theorem
References
Section 2.3
Maximum Value
Local Maximum
T or F
An absolute max is a local max.
The Extreme Value Theorem
Fermat’s Theorem
Lemma (HW)
Fermat’s Theorem
Conceptual Diagrams
Fermat’s Theorem
Fermat’s Theorem
Fermat’s Theorem
Fermat’s Theorem
Proof
Proof
Proof
Proof
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Rolle’s Theorem
Proof
Proof
The Mean Value Theorem
Proof
The Mean Value Theorem
The Mean Value Theorem
The Mean Value Theorem
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?
Corollary (Important)
)()( xgxf Cxgxf )()(
a b
)(xfy
)(xgy
Corollary (Important)
)()( xgxf Cxgxf )()(
a b
)(xfy
)(xgy C