Paper 1, Section II, E

Analysis I | Part IA, 2013

(i) State (without proof) Rolle's Theorem.

(ii) State and prove the Mean Value Theorem.

(iii) Let f,g:[a,b]Rf, g:[a, b] \rightarrow \mathbb{R} be continuous, and differentiable on (a,b)(a, b) with g(x)0g^{\prime}(x) \neq 0 for all x(a,b)x \in(a, b). Show that there exists ξ(a,b)\xi \in(a, b) such that


Deduce that if moreover f(a)=g(a)=0f(a)=g(a)=0, and the limit

=limxaf(x)g(x)\ell=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}

exists, then

f(x)g(x) as xa\frac{f(x)}{g(x)} \rightarrow \ell \text { as } x \rightarrow a

(iv) Deduce that if f:RRf: \mathbb{R} \rightarrow \mathbb{R} is twice differentiable then for any aRa \in \mathbb{R}

f(a)=limh0f(a+h)+f(ah)2f(a)h2.f^{\prime \prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} .

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