In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$\frac{- 2^{x} \log{\left(2 \right)} + 2^{- x} \log{\left(2 \right)}}{2^{x} + 2^{- x}} + \frac{2^{x} \log{\left(2 \right)} - 2^{- x} \log{\left(2 \right)}}{2^{x} + 2^{- x}} = 0$$
Solve this equationSolutions are not found,
function may have no extrema