Given function for $x>0$
$$f(x)=(1+\frac1x)^x(1+x)^\frac1x$$
which is not a monotonic function, but it is easy to find the only maxima
$$f(1)=4$$
so, can we find a strict prove showing $f(x)$ is monotonic increasing for interval $(0,1)$, and monotonic decreasing for interval $(1,\infty)$?
actually, I already try to do the logarithm of the function shows
$$\ln f(x)=x\ln(1+\frac1x)+\frac1x\ln(1+x)$$
but, it still seems not convenient to reach the monotonicity, even take the derivative of $\ln f(x)$, especially for interval $(0,1)$. I may need some further suggestions here. Thanks for any help.
Hint: Write your $f(x)$ in the form $$f(x)=\frac{(1+x)^x\cdot (1+x)^{1/x}}{x^x}$$ and take the logarithm of your $f(x)$ $$\ln(f(x))=\left(x+\frac{1}{x}\right)\ln(x+1)-x\ln(x)$$