Introduction to Mathematical Analysis I - Second Edition

126 4.6 CONVEX FUNCTIONS AND DERIVATIVES Exercises 4.5.1 B Use Taylor’s theorem to prove that e x > m ∑ k = 0 x k k ! for all x > 0 and m ∈ N . 4.5.2 Find the 5th Taylor polynomial, P 5 ( x ) , at ¯ x = 0 for cos x . Determine an upper bound for the error | P 5 ( x ) − cos x | for x ∈ [ − π / 2 , π / 2 ] . 4.5.3 Use Theorem 4.5.3 to determine if the following functions have a local minimum or a local maximum at the indicated points. (a) f ( x ) = x 3 sin x at ¯ x = 0. (b) f ( x ) = ( 1 − x ) ln x at ¯ x = 1. 4.5.4 Suppose f is twice differentiable on ( a , b ) . Show that for every x ∈ ( a , b ) , lim h → 0 f ( x + h )+ f ( x − h ) − 2 f ( x ) h 2 = f 00 ( x ) . 4.5.5 I (a) Suppose f is three times differentiable on ( a , b ) and ¯ x ∈ ( a , b ) . Prove that lim h → 0 f ( ¯ x + h ) − f ( ¯ x ) − f 0 ( ¯ x ) h 1! − f 00 ( ¯ x ) h 2 2! h 3 = f 000 ( ¯ x ) 3! . (b) State and prove a more general result for the case where f is n times differentiable on ( a , b ) . 4.5.6 Suppose f is n times differentiable on ( a , b ) and ¯ x ∈ ( a , b ) . Define P n ( h ) = n ∑ k = 0 f ( n ) ( ¯ x ) h n n ! for h ∈ R Prove that lim h → 0 f ( ¯ x + h ) − P n ( h ) h n = 0 . (Thus, we have f ( ¯ x + h ) = P n ( h )+ g ( h ) , wher g is a function that satisfies lim h → 0 g ( h ) h n = 0. This is called the Taylor expansion with Peano’s remainder .) 4.6 CONVEX FUNCTIONS AND DERIVATIVES We discuss in this section a class of functions that plays an important role in optimization problems.

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