Analysis | Part IA, 2004

i) State Rolle's theorem.

Let f,g:[a,b]Rf, g:[a, b] \rightarrow \mathbb{R} be continuous functions which are differentiable on (a,b)(a, b).

ii) Prove that for some c(a,b)c \in(a, b),

(f(b)f(a))g(c)=(g(b)g(a))f(c).(f(b)-f(a)) g^{\prime}(c)=(g(b)-g(a)) f^{\prime}(c) .

iii) Suppose that f(a)=g(a)=0f(a)=g(a)=0, and that limxa+f(x)g(x)\lim _{x \rightarrow a+} \frac{f^{\prime}(x)}{g^{\prime}(x)} exists and is equal to LL.

Prove that limxa+f(x)g(x)\lim _{x \rightarrow a+} \frac{f(x)}{g(x)} exists and is also equal to LL.

[You may assume there exists a δ>0\delta>0 such that, for all x(a,a+δ),g(x)0x \in(a, a+\delta), g^{\prime}(x) \neq 0 and g(x)0.]g(x) \neq 0 .]

iv) Evaluate limx0logcosxx2\lim _{x \rightarrow 0} \frac{\log \cos x}{x^{2}}.

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