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 nbe 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 nbe 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)and limx→∞ 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?
RkJQdWJsaXNoZXIy NTc4NTAz