Introduction to Mathematical Analysis I - Second Edition

112 4.2 THE MEAN VALUE THEOREM Figure 4.4: Right derivative. Proof: Define the function g : [ a , b ] → R by g ( x ) = f ( x ) − λ x . Then g is differentiable on [ a , b ] and g 0 + ( a ) < 0 < g 0 − ( b ) . Thus, lim x → a + g ( x ) − g ( a ) x − a < 0 . It follows that there exists δ 1 > 0 such that g ( x ) < g ( a ) for all x ∈ ( a , a + δ 1 ) ∩ [ a , b ] . Similarly, there exists δ 2 > 0 such that g ( x ) < g ( b ) for all x ∈ ( b − δ 2 , b ) ∩ [ a , b ] . Since g is continuous on [ a , b ] , it attains its minimum at a point c ∈ [ a , b ] . From the observations above, it follows that c ∈ ( a , b ) . This implies g 0 ( c ) = 0 or, equivalently, that f 0 ( c ) = λ . Remark 4.2.6 The same conclusion follows if f 0 + ( a ) > λ > f 0− ( b ) .