Introduction to Mathematical Analysis I - Second Edition

113 Exercises 4.2.1 B Let f and g be differentiable at x 0 . Suppose f ( x 0 ) = g ( x 0 ) and f ( x ) ≤ g ( x ) for all x ∈ R . Prove that f 0 ( x 0 ) = g 0 ( x 0 ) . 4.2.2 Prove the following inequalities using the Mean Value Theorem. (a) √ 1 + x < 1 + 1 2 x for x > 0. (b) e x > 1 + x , for x > 0. (Assume known that the derivative of e x is itself.) (c) x − 1 x < ln x < x − 1, for x > 1. (Assume known that the derivative of ln x is 1 / x .) 4.2.3 I Prove that | sin ( x ) − sin ( y ) | ≤ | x − y | for all x , y ∈ R . 4.2.4 B Let n be a positive integer and let a k , b k ∈ R for k = 1 , . . . , n . Prove that the equation x + n ∑ k = 1 ( a k sin kx + b k cos kx ) = 0 has a solution on ( − π , π ) . 4.2.5 B Let f and g be differentiable functions on [ a , b ] . Suppose g ( x ) 6 = 0 and g 0 ( x ) 6 = 0 for all x ∈ [ a , b ] . Prove that there exists c ∈ ( a , b ) such that 1 g ( b ) − g ( a ) f ( a ) f ( b ) g ( a ) g ( b ) = 1 g 0 ( c ) f ( c ) g ( c ) f 0 ( c ) g 0 ( c ) , where the bars denote determinants of the two-by-two matrices. 4.2.6 B L et n be a fixed positive integer. (a) Suppose a 1 , a 2 , . . . , a n satisfy a 1 + a 2 2 + · · · + a n n = 0 . Prove that the equation a 1 + a 2 x + a 3 x 2 + · · · + a n x n − 1 = 0 has a solution in ( 0 , 1 ) . (b) Suppose a 0 , a 1 , . . . , a n satisfy n ∑ k = 0 a k 2 k + 1 = 0 .