maximum of $\frac{1}{(1+(x- \frac{1}{2})^2 )^{\frac{3}{2}}}+\frac{1}{(1+(x+ \frac{1}{2})^2 )^{\frac{3}{2}}}$ How to find this function's maximum?
$$\frac{1}{(1+(x- \frac{1}{2})^2 )^{\frac{3}{2}}}+\frac{1}{(1+(x+ \frac{1}{2})^2 )^{\frac{3}{2}}}$$
I think it has a maximum value at x=0. A search on Wolfram Alpha revealed that this is correct. However, the result in wolframalpha was to solve the 12th-order equation by differentiating and then performing general differentiation, I wonder if there is an easier way.

Hint :
Let :
$$f(x)=\frac{1}{(1+(x- \frac{1}{2})^2 )^{\frac{3}{2}}}+\frac{1}{(1+(x+ \frac{1}{2})^2 )^{\frac{3}{2}}}$$
We find the antiderivative of $f(x)$ wich is :
$$F(x)=g(x)-g(-x)$$
Where :
$$\frac{2\left(x+1\right)}{\sqrt{4x^{2}+2x+5}}=g(x)$$
Now we want to show that :
$$h(x)=F(x)-\frac{16}{5\sqrt{5}}x$$
Is decreasing on $x\in[0,\infty)$
The trick is to split in two the problem introducing a new function such that :
$$u(x)=g(x)-\frac{8}{5\sqrt{5}}\left(x+\frac{a}{x^{2}+1}\right)$$
And :
$$v(x)=-g(-x)-\frac{8}{5\sqrt{5}}\left(x-\frac{a}{x^{2}+1}\right)$$
Where $a\simeq 0.674$ and such that the two functions $u(x)$ and $v(x)$ are decreasing and admits a single extrema at $x=0$
The result follows .
