Maxima and minima — using calculus — for IITJEE Advanced Mathematics

Problem: 

The length of the edge of the cube ABCDA_{1}B_{1}C_{1}D_{1} is 1. Two points M and N move along the line segments AB and A_{1}D_{1}, respectively, in such a way that at any time t (0 \leq t < \infty) we have BM = |\sin{t}| and D_{1}N=|\sin(\sqrt{2}t)|. Show that MN has no maximum.

Proof:

Clearly, MN \geq MA_{1} \geq AA_{1} =1. If MN=1 for some t, then M=A and N=A_{1}, which is equivalent to |\sin{t}|=1 and |\sin{\sqrt{2}t}|=1. Consequently, t=\frac{\pi}{2}+k\pi and \sqrt{2}t=\frac{\pi}{2}+n\pi for some integers k and n, which implies \sqrt{2}=\frac{2n+1}{2k+1}, a contradiction since \sqrt{2} is irrational. This is why MN >1 for any t.

We will now show that MN can be made arbitrarily close to 1. For any integer k, set t_{k}=\frac{\pi}{2}+k\pi. Then, |\sin{t_{k}}|=1, so at any time t_{k}, the point M is at A. To show that N can be arbitrarily close to A_{1} at times t_{k}, it is enough to show that |\sin(\sqrt{2}t_{k})| can be arbitrarily close to 1 for appropriate choices of k.

We are now going to use Kronecker’s theorem: 

If \alpha is an irrational number, then the set of numbers of the form ma+nwhere m is a positive integer while n is an arbitrary integer, is dense in the set of all real numbers. The latter means that every non-empty open interval (regardless of how small it is) contains a number of the form ma+n.

Since \sqrt{2} is irrational, we can use Kronecker’s theorem with \alpha=\sqrt{2}. Then, for x=\frac{1-\sqrt{2}}{2}, and any \delta>0, there exist integer k \geq 1 and n_{k} such  that k\sqrt{2}-n_{k} \in (x-\delta, x+\delta). That is, for \in_{k}=\sqrt{2}k+\frac{\sqrt{2}}{2}-\frac{1}{2}-n_{k} we have |\in_{k}|<\delta. Since \sqrt{2}(k+\frac{1}{2})=\frac{1}{2}+n_{k}+\in_{k}, we have

|\sin(\sqrt{2}t_{k})|=|\sin \pi \sqrt{2}(k+\frac{1}{2})|=|\sin(\frac{\pi}{2}+n_{k}\pi+\in_{k}\pi)|=|\cos{(\pi \in_{k})}|.

It remains to note that |\cos{(\delta\pi)}| tends to 1 as \delta tends to 0.

Hence, MN can be made arbitrarily close to 1.

Ref: Geometric Problems on Maxima and Minima by Titu Andreescu, Oleg Mushkarov, and Luchezar Stoyanov.

Thanks to Prof Andreescu, et al !

Nalin Pithwa

 

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