Introduction to Mathematical Analysis I - Second Edition

109 Figure 4.2: Illustration of Rolle’s Theorem. If ¯ x 1 ∈ ( a , b ) or ¯ x 2 ∈ ( a , b ) , then f has a local minimum at ¯ x 1 or f has a local maximum at ¯ x 2 . By Theorem 4.2.1 , f 0 ( ¯ x 1 ) = 0 or f 0 ( ¯ x 2 ) = 0, and ( 4.3 ) holds with c = ¯ x 1 or c = ¯ x 2 . If both ¯ x 1 and ¯ x 2 are the endpoints of [ a , b ] , then f ( ¯ x 1 ) = f ( ¯ x 2 ) because f ( a ) = f ( b ) . By ( 4.4 ) , f is a constant function, so f 0 ( c ) = 0 for any c ∈ ( a , b ) . We are now ready to use Rolle’s Theorem to prove the Mean Value Theorem presented below. Figure 4.3: Illustration of the Mean Value Theorem. Theorem 4.2.3 — Mean Value Theorem. Let a , b ∈ R with a < b and f : [ a , b ] → R . Suppose f is continuous on [ a , b ] and differentiable on ( a , b ) . Then there exists c ∈ ( a , b ) such that f 0 ( c ) = f ( b ) − f ( a ) b − a . (4.5) Proof: The linear function whose graph goes through ( a , f ( a )) and ( b , f ( b )) is g ( x ) = f ( b ) − f ( a ) b − a ( x − a )+ f ( a ) .