Paper 1, Section II, E

Analysis I | Part IA, 2016

State the Bolzano-Weierstrass theorem. Use it to show that a continuous function f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} attains a global maximum; that is, there is a real number c[a,b]c \in[a, b] such that f(c)f(x)f(c) \geqslant f(x) for all x[a,b]x \in[a, b].

A function ff is said to attain a local maximum at cRc \in \mathbb{R} if there is some ε>0\varepsilon>0 such that f(c)f(x)f(c) \geqslant f(x) whenever xc<ε|x-c|<\varepsilon. Suppose that f:RRf: \mathbb{R} \rightarrow \mathbb{R} is twice differentiable, and that f(x)<0f^{\prime \prime}(x)<0 for all xRx \in \mathbb{R}. Show that there is at most one cRc \in \mathbb{R} at which ff attains a local maximum.

If there is a constant K<0K<0 such that f(x)<Kf^{\prime \prime}(x)<K for all xRx \in \mathbb{R}, show that ff attains a global maximum. [Hint: if g(x)<0g^{\prime}(x)<0 for all xRx \in \mathbb{R}, then gg is decreasing.]

Must f:RRf: \mathbb{R} \rightarrow \mathbb{R} attain a global maximum if we merely require f(x)<0f^{\prime \prime}(x)<0 for all xR?x \in \mathbb{R} ? Justify your answer.

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