In interpolation theory, given a compatible couple of Banach spaces $(X_0, X_1)$ one considers the $J$ and $K$-functionals, defined as follows:

If $x \in X_0 + X_1$ and $t > 0$ then $$K(t, x) = \inf\{\|x_0\|_{X_0} + t\|x_1\|_{X_1} : x = x_0 + x_1, x_0 \in X_0, x_1 \in X_1\}.$$

If $x \in X_0 \cap X_1$ and $t > 0$ then $$J(t, x) = \max\{\|x\|_{X_0}, t\|x\|_{X_1}\}.$$

The books I checked gave those definitions without explaining where they came from. What is the motivation for defining the $J$ and $K$-functionals?