Paper 1, Section II, E

Analysis | Part IA, 2007

Let a<ba<b be real numbers, and let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be continuous. Show that ff is bounded on [a,b][a, b], and that there exist c,d[a,b]c, d \in[a, b] such that for all x[a,b]x \in[a, b], f(c)f(x)f(d)f(c) \leqslant f(x) \leqslant f(d).

Let g:RRg: \mathbb{R} \rightarrow \mathbb{R} be a continuous function such that

limx+g(x)=limxg(x)=0\lim _{x \rightarrow+\infty} g(x)=\lim _{x \rightarrow-\infty} g(x)=0

Show that gg is bounded. Show also that, if aa and cc are real numbers with 0<cg(a)0<c \leqslant g(a), then there exists xRx \in \mathbb{R} with g(x)=cg(x)=c.

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