Paper 1, Section II, D

Analysis I | Part IA, 2009

State and prove Rolle's theorem.

Let ff and gg be two continuous, real-valued functions on a closed, bounded interval [a,b][a, b] that are differentiable on the open interval (a,b)(a, b). By considering the determinant

ϕ(x)=110f(a)f(b)f(x)g(a)g(b)g(x)=g(x)(f(b)f(a))f(x)(g(b)g(a))\phi(x)=\left|\begin{array}{ccc} 1 & 1 & 0 \\ f(a) & f(b) & f(x) \\ g(a) & g(b) & g(x) \end{array}\right|=g(x)(f(b)-f(a))-f(x)(g(b)-g(a))

or otherwise, show that there is a point c(a,b)c \in(a, b) with


Suppose that f,g:(0,)Rf, g:(0, \infty) \rightarrow \mathbb{R} are differentiable functions with f(x)0f(x) \rightarrow 0 and g(x)0g(x) \rightarrow 0 as x0x \rightarrow 0. Prove carefully that if the limitx0f(x)g(x)=\operatorname{limit}_{x \rightarrow 0} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\ell exists and is finite, then the limit limx0f(x)g(x)\lim _{x \rightarrow 0} \frac{f(x)}{g(x)} also exists and equals \ell.

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