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Prove this inequality $x^{1-x\ }+\log \left(1-x\right)\ +2^{x\ }+\left(1-x\right)^x+\log \left(x\right)+2^{\left(1-x\right)}<3.7$

I came up with this on my own and checked this using Desmos. Will edit in my attempts after a few days.

My Thoughts

  1. Differentiating the function. This is bound to give us an answer. But the main problem I feel here is that it will be difficult to get a analytic solution. This method will also take up time and is pretty cumbersome.

  2. This is what I'm specifically looking for, a non calculus solution!

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    $\begingroup$ What does the derivative tell you? This function is symmetric about $x=\frac{1}{2}$ (and only defined in $[0,1]$). $\endgroup$ – Michael Burr Mar 16 '17 at 14:38
  • $\begingroup$ @Michael Burr...It is not defined at 1.. $\endgroup$ – user35508 Mar 16 '17 at 14:45
  • $\begingroup$ @user35508. The main point is the symmetry pointed out by Michael Burr; from this we can immediately conclude. I wonder why $3.7$ and not less. $\endgroup$ – Claude Leibovici Mar 16 '17 at 14:50
  • $\begingroup$ @ClaudeLeibovici Pls can you give your method in a answer. Thanks:) $\endgroup$ – Agile_Eagle Mar 16 '17 at 15:07
  • $\begingroup$ Michael Burr gave already the answer. The maximum of the function is at $x=\frac 12$. Plug it in the lhs to get a maximum value $3 \sqrt{2}-2\log (2)\approx 2.85635$ $\endgroup$ – Claude Leibovici Mar 16 '17 at 15:13

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