Paper 1, Section II, E

Analysis | Part IA, 2007

State and prove the Mean Value Theorem.

Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be a function such that, for every xR,f(x)x \in \mathbb{R}, f^{\prime \prime}(x) exists and is non-negative.

(i) Show that if xyx \leqslant y then f(x)f(y)f^{\prime}(x) \leqslant f^{\prime}(y).

(ii) Let λ(0,1)\lambda \in(0,1) and a<ba<b. Show that there exist xx and yy such that

f(λa+(1λ)b)=f(a)+(1λ)(ba)f(x)=f(b)λ(ba)f(y)f(\lambda a+(1-\lambda) b)=f(a)+(1-\lambda)(b-a) f^{\prime}(x)=f(b)-\lambda(b-a) f^{\prime}(y)

and that

f(λa+(1λ)b)λf(a)+(1λ)f(b).f(\lambda a+(1-\lambda) b) \leqslant \lambda f(a)+(1-\lambda) f(b) .

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