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If $x_1,x_2...x_n$ are postive numbers satisfying $x_1\cdot x_2 \cdots x_n = 1$ , then find the maximum value of $\frac{1}{\sqrt{x_1 ^2 + 1}\,\cdot\, \sqrt{x_2^2 +1}\,\cdots\,\sqrt{x_n^2+1} }$ .

What i considered was applying the RMS -GM inequality which is $\sqrt{a^2 +b^2} \geq 2\sqrt{ab}$ which gives the maximum of the expression but is there an another way which can be used to systematically show and get the maximum ?

  • Note : originally problem was given in terms of $\cot z_1 \cdots \cot z_n = 1$ and all angles lie in $(0,π/2)$ and we need to find maximum of $\cos z_1\cdots\cos z_n$ but for better understanding i converted into this form
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  • $\begingroup$ The problem has a full symmetry over variables $x_1,...., x_n$. It means that we can check the case $x_1=x_2=... =x_n$ first. We can see that it is the maximum. Indeed, let's take $x_1=x_2=...=x_{n-2}=1$ and $x_{n-1}=\frac{1}{x_n}$; then, leading $x_n\to 0$, we can get as small value as we want. $\endgroup$
    – Svyatoslav
    Jun 24, 2022 at 17:52
  • $\begingroup$ I understood most but how just checxking the x_1 = x_2 = ... Case one concludes its the maximum ? @Svyatoslav $\endgroup$ Jun 24, 2022 at 18:00

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For the rigorous proof it is convenient to use the method of Lagrange multipliers https://en.wikipedia.org/wiki/Lagrange_multiplier#:~:text=In%20mathematical%20optimization%2C%20the%20method,chosen%20values%20of%20the%20variables).

We can define $\displaystyle f(x_1, ..., x_n)=\ln \frac{1}{\sqrt{x_1 ^2 + 1}\,\cdot\, \sqrt{x_2^2 +1}\,\cdots\,\sqrt{x_n^2+1} }=-\frac{1}{2}\Big(\ln(x_1^2+1)+...+\ln(x_n^2+1)\Big)$.

As logarithm is a growing function, it is enough to find the maximum of $f(x)$. Given that $\ln(x_1...x_n)=0$, we also introduce the additional condition via $g(x_1, ..., x_n)=\ln(x_1...x_n)=\ln x_1+... \ln x_n$, and consider the Lagrangian function $$L(x_1, ... x_n)=f(x_1, ..., x_n)+\lambda g(x_1, ..., x_n)$$ We we are looking for an unconditional extremum of $L(x_1, ... x_n)$: $$\frac{\partial}{\partial x_i}L(x_1, ... x_n)=0\,\,\Rightarrow\,\,\frac{\lambda}{x_i}-\frac{x_i}{x_i^2+1}=0\,\,\Rightarrow\,\,x_i^2=\frac{1-\lambda}{\lambda}$$ From the condition $\displaystyle g(x_1, ..., x_n)=\ln x_1+... \ln x_n=0$ we get $$2n\ln\frac{1-\lambda}{\lambda}=0\,\,\Rightarrow\,\,\lambda=\frac{1}{2}\,\,\Rightarrow\,\, \,x_1=x_2=...=x_n=1$$ We have a single conditional extremum; taking, for example, $x_1=...x_{n-2}=1,\,x_{n-1}=\frac{1}{x_n}$ and leading $x_n\to 0$ (and $x_{n-1}\to\infty$ correspondingly), we can make $f(x)$ as small as we wish; therefore, this extremum is a maximum.

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