101 (a) √1+x <1+1 2x for x >0. (b) ex >1+x, for x >0. (Assume known that the derivative of ex is itself.) (c) x−1 x <lnx <x−1, for x >1. (Assume known that the derivative of lnx is 1/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 ∈Rfor k =1,...,n. Prove that the equation x+ n ∑k=1 (ak sinkx+bk coskx)=0 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 g(b)−g(a) f (a) f (b) g(a) g(b) = 1 g′(c) f (c) g(c) f ′(c) g′(c) , where the bars denote determinants of the two-by-two matrices. 4.2.6 ▷Let n be a fixed positive integer. (a) Suppose a1,a2,...,an satisfy a1 + a2 2 +···+ an n =0. Prove that the equationa1 +a2x+a3x 2 +···+anx n−1 =0 has a solution in(0,1). (b) Suppose a0,a1,...,an satisfy n ∑k=0 ak 2k+1 =0. Prove that the equation∑n k=0ak cos(2k+1)x =0 has a solution on(0, π 2). 4.2.7 Let f : [0,∞)→Rbe a differentiable function. Prove that if both limx →∞ f (x)andlimx →∞ f ′(x) exist, then limx →∞ f ′(x)=0 4.2.8 ▷Let f : [0,∞) →Rbe a differentiable function. (a) Show that if limx →∞ f ′(x)=a, then limx →∞ f (x) x =a. (b) Show that if limx →∞ f ′(x)=∞, then limx →∞ f (x) x =∞. (c) Are the converses in part (a) and part (b) true?
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