I'm interested by the following problem :
Let $a,b,c>0$ such that $abc=a+b+c$ and $a\geq b \geq c$ then we have : $$f(a,b,c)=\exp\Big(\frac{1}{7a+b}\Big)+\exp\Big(\frac{1}{7b+c}\Big)+\exp\Big(\frac{1}{7c+a}\Big)\leq 3\exp\Big(\frac{1}{8\sqrt{3}}\Big)$$
My try : I transform this inequality in a problem of tangent line on the exponential (because $e^x=(e^x)'$) and with the condition we have (it's checked) : $$f(a,b,c)\leq\sum_{cyc} \frac{\exp\Big(\frac{1}{7a+b}\Big)-\exp\Big(\frac{1}{8\sqrt{3}}\Big)}{\frac{1}{7a+b}-\frac{1}{8\sqrt{3}}}\leq 3\exp\Big(\frac{1}{8\sqrt{3}}\Big)$$
After this I want to use the three chord lemma but it gives not enought informations .
Maybe we can use the Lagrange multiplier but I don't control this method .
If you have hints it would be nice and thanks a lot for your time .
Ps: This is the exponential twin of this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$