Compare the following numbers: $$a=2016^{\sqrt{2014}}, b=2015^{\sqrt{2015}},c=2014^\sqrt{2016}$$ The answer to it was the following: $$f(x) = \sqrt{x+1}\ln(x-1)- \sqrt{x}\ln x$$ defined for any $$x\in[2;\infty)$$the function has a local maximum, strictly positive in a point with the abscissa $x_1\in(62;63)$, $$\lim_{x\to\infty}f(x)=0$$ and is strictly decreasing on the interval $[x_1;\infty)$, so it's also strictly positive on the interval. Applying the exponential we get $f(2015)>0$ which is $c>b$.
The function $$g(x) = \sqrt{x}\ln x - \sqrt{x-1}\ln(x+1)$$, defined for any $$x\in[2;\infty)$$ has a strict positive local maximum in a poit of abscisa $x_2\in(45;46)$, $$\lim_{x\to\infty}g(x)=0$$ and it's strictly decreasing on the interval $[x_2;\infty)$, so it's also positive on this interval. So $f(2016)>0$ which means $b>a$.
So c>b>a.
I see the logic of it. But my question is how on earth did the author find $x_1$ and $x_2$, the two local maximums of the functions? I've calculated the derivative and it's horrible. Please give some light! Thank you!