MAT 3749Introduction to Analysis

Section 2.3 Part III

The Mean Value Theorem

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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