Introduction to Mathematical Analysis I - Second Edition

125 (a) f ( n ) ( ¯ x ) > 0 if and only if f has a local minimum at ¯ x . (b) f ( n ) ( ¯ x ) < 0 if and only if f has a local maximum at ¯ x . Proof: We will prove (a) . Suppose f ( n ) ( ¯ x ) > 0. Since f ( n ) ( ¯ x ) > 0 and f ( n ) is continuous at ¯ x , there exists δ > 0 such that f ( n ) ( t ) > 0 for all t ∈ B ( ¯ x ; δ ) ⊂ ( a , b ) . Fix any x ∈ B ( ¯ x ; δ ) . By Taylor’s theorem and the given assumption, there exists c in between ¯ x and x such that f ( x ) = f ( ¯ x )+ f ( n ) ( c ) n ! ( x − ¯ x ) n . Since n is even and c ∈ B ( ¯ x ; δ ) , we have f ( x ) ≥ f ( ¯ x ) . Thus, f has a local minimum at ¯ x . Now, for the converse, suppose that f has a local minimum at ¯ x . Then there exists δ > 0 such that f ( x ) ≥ f ( ¯ x ) for all x ∈ B ( ¯ x ; δ ) ⊂ ( a , b ) . Fix a sequence { x k } in ( a , b ) that converges to ¯ x with x k 6 = ¯ x for every k . By Taylor’s theorem , there exists a sequence { c k } , with c k between x k and ¯ x for each k , such that f ( x k ) = f ( ¯ x )+ f ( n ) ( c k ) n ! ( x k − ¯ x ) n . Since x k ∈ B ( ¯ x ; δ ) for sufficiently large k , we have f ( x k ) ≥ f ( ¯ x ) for such k . It follows that f ( x k ) − f ( ¯ x ) = f ( n ) ( c k ) n ! ( x k − ¯ x ) n ≥ 0 . This implies f ( n ) ( c k ) ≥ 0 for such k . Since { c k } converges to ¯ x , f ( n ) ( ¯ x ) = lim k → ∞ f ( n ) ( c k ) ≥ 0. The proof of (b) is similar. Example 4.5.2 Consider the function f ( x ) = x 2 cos x defined on R . Then f 0 ( x ) = 2 x cos x − x 2 sin x and f 00 ( x ) = 2cos x − 4 x sin x − x 2 cos x . Then f ( 0 ) = f 0 ( 0 ) = 0 and f 00 ( 0 ) = 2 > 0. It follows from the previous theorem that f has a local minimum at 0. Notice, by the way, that since f ( 0 ) = 0 and f ( π ) < 0, 0 is not a global minimum. Example 4.5.3 Consider the function f ( x ) = − x 6 + 2 x 5 + x 4 − 4 x 3 + x 2 + 2 x − 3 defined on R . A direct calculations shows f 0 ( 1 ) = f 00 ( 1 ) = f 000 ( 1 ) = f ( 4 ) ( 1 ) = 0 and f ( 5 ) ( 1 ) < 0. It follows from the previous theorem that f has a local maximum at 1.