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What I tried was pretty basic.

$$A=2xy=2x\cos x$$

$$A'=2\cos x-2x\sin x$$

$$A' = 0 \implies 2\cos x-2x\sin x=0$$

$$x=\frac{\cos x}{\sin x}=\cot x$$

But I got stuck here. I didn't know what to do next.

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Eyad H.
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  • Analytical solution is pretty far fetched. You'll have to settle for a numerical one, say, a Newton's Approximation. – Jack Tiger Lam Dec 14 '16 at 22:56
  • @JackLam I don't know what a Newton's approximation is. I am a senior secondary school student and this was a question we were asked to solve (before they told us not to anyways) with our knowledge with local/absolute maxima and minima. Was the problem wrongly placed? I mean, not understanding what you said, I feel like it isn't for us and is more advanced? – Eyad H. Dec 14 '16 at 22:59
  • @EyadH. before they told us not to anyways That's probably when someone realized that there is no "nice" closed form solution. You can prove that $x = \cot x$ has a unique solution in $(0,\pi / 2)$ and that value gives the maximum area, but that's about as far as you can push it symbolically i.e. without solving the equation numerically. – dxiv Dec 14 '16 at 23:07

1 Answers1

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You cannot express the solution of this equation by means of the usual functions, and you need to resort to so-called numerical methods that let you evaluate approximate values.

Consider the sequence of numbers

$$x_{n+1}=\arctan\frac1{x_n},$$ starting from $x_0=1$. If these values get closer and closer to a certain $x$, then you can say that "in the end",

$$x=\arctan\frac1x,$$ which is the same as $x=\cot x$. Have a try with a calculator.

You will later learn under what conditions such a process can converge to the solution of an equation (and why I chose this particular form of recurrence). You will also learn faster methods.