104 4.4 L’Hôpital’s Rule Exercises 4.3.1 (a) Let f : R→Rbe differentiable. Prove that if f ′(x) is bounded, then f is Lipschitz continuous and, in particular, uniformly continuous. (b) Give an example of a function f : (0,∞) →Rwhich is differentiable and uniformly continuous but such that f ′(x) is not bounded. 4.3.2 ▶Let f : R→R. Suppose there exist ℓ ≥0 and α>0 such that | f (u)−f (v)| ≤ℓ|u−v|α for all u,v ∈R. (4.8) (a) Prove that f is uniformly continuous onR. (b) Prove that if α>1, then f is a constant function. (c) Find a nondifferentiable function that satisfies the condition above for α=1. 4.3.3 ▷Let f andgbe differentiable functions onRsuch that f (x0) =g(x0) and f ′(x) ≤g′(x) for all x ≥x0. Prove that f (x) ≤g(x) for all x ≥x0. 4.3.4 Let f,g: R→Rbe differentiable functions satisfying (a) f (0) =g(0) =1 (b) f (x) >0, g(x) >0 and f ′(x) f (x) > g′(x) g(x) for all x. Prove that f (1) g(1) >1> g(1) f (1) . 4.3.5 ▷Let f be twice differentiable on an open interval I. Suppose that there exist a,b,c ∈I with a<b<c such that f (a) <f (b) and f (b) >f (c). Prove that there exists d ∈(a,c) such that f ′′(d) <0. 4.3.6 ▷Prove that the function f defined in Exercise 4.1.11 is not monotone on any open interval containing 0. 4.4 L’Hôpital’s Rule We now prove a result that allows us to compute various limits by calculating a related limit involving derivatives. All four theorems in this section are known as L’Hôpital’s Rule. In this section, we assume a,b∈Rwitha<b. Theorem 4.4.1 Let f andgbe continuous on [a,b] and differentiable on (a,b). Suppose that: (i) f (x0) =g(x0) =0, where x0 ∈[a,b], (ii) there exists δ >0 such that g′(x)̸ =0 for all x ∈(x0 −δ,x0 +δ)∩[a,b], x̸ =x0.
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