101 √ 1 (a) 1 + x < 1 + 2 x for x > 0. (b) ex > 1+ x, for x > 0. (Assume known that the derivative of ex is itself.) x− 1 (c) < lnx < x − 1, for x > 1. (Assume known that the derivative of lnx is 1/x.) x 4.2.3 ▶ Prove that |sin(x) − sin(y)|≤|x − y| for all x,y ∈ R. 4.2.4 ▷ Let n be a positive integer and let ak,bk ∈ R for k = 1,...,n. Prove that the equation n x+ ∑(ak sinkx + bk coskx)= 0 k=1 has a solution on (−π,π). 4.2.5 ▷ Let f and g be differentiable functions on [a,b]. Suppose g(x)≠ 0 and g ′ (x)≠ 0 for all x ∈ [a,b]. Prove that there exists c ∈ (a,b) such that 1 f (a) f (b) 1 f (c) g(c) = , g(b) − g(a) g(a) g(b) g ′ (c) f ′ (c) g ′ (c) where the bars denote determinants of the two-by-two matrices. 4.2.6 ▷Let n be a fxed positive integer. (a) Suppose a1,a2,...,an satisfy a2 an a1 + + ··· + = 0. 2 n 2 + anx n−1 Prove that the equation a1 + a2x + a3x + ··· = 0 has a solution in (0,1). (b) Suppose a0,a1,...,an satisfy ∑ n ak = 0. 2k + 1 k=0 Prove that the equation ∑n k=0 ak cos(2k + 1)x = 0 has a solution on (0, π 2 ). 4.2.7 Let f : [0,∞) → R be a differentiable function. Prove that if both limx →∞ f (x) and limx →∞ f ′ (x) exist, then limx →∞ f ′ (x)= 0 4.2.8 ▷ Let f : [0,∞) → R be a differentiable function. f (x) (a) Show that if limx →∞ f ′ (x)= a, then limx →∞ = a. x f (x) (b) Show that if limx →∞ f ′ (x)= ∞, then limx →∞ = ∞. x (c) Are the converses in part (a) and part (b) true?
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