We give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions
$f_\alpha:[0,1]\to{\mathbb R}$
parameterised by
$\alpha\in(-2,2)$
. We show that
$f_\alpha$
has a unique maximiser in
$[0,1/2]$
if and only if there does not exist a Littlewood polynomial that has
$\alpha$
as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with
$\alpha\in(-2,1/2]\cup(1,2)$
. For
$(1/2,1]$
, we show that the step roots are dense in that interval. If
$\alpha\in (1/2,1]$
is a step root, then the set of maximisers of
$f_\alpha$
is an explicitly given perfect set with Hausdorff dimension
$1/(n+1)$
, where n is the degree of the minimal Littlewood polynomial that has
$\alpha$
as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to
$[-2,-1/2]\cup[1/2,2]$
.