Paper 1, Section II, 11D11 D

Analysis I | Part IA, 2015

(i) State and prove the intermediate value theorem.

(ii) Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be a continuous function. The chord joining the points (α,f(α))(\alpha, f(\alpha)) and (β,f(β))(\beta, f(\beta)) of the curve y=f(x)y=f(x) is said to be horizontal if f(α)=f(β)f(\alpha)=f(\beta). Suppose that the chord joining the points (0,f(0))(0, f(0)) and (1,f(1))(1, f(1)) is horizontal. By considering the function gg defined on [0,12]\left[0, \frac{1}{2}\right] by


or otherwise, show that the curve y=f(x)y=f(x) has a horizontal chord of length 12\frac{1}{2} in [0,1][0,1]. Show, more generally, that it has a horizontal chord of length 1n\frac{1}{n} for each positive integer nn.

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