  The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding
for every pair of vertices of $G$, which is not covered by an edge in $G$, an
extra edge containing this pair and $(r-2)$ new vertices. In this paper we
determine the Tur\'an number of the extension of an $r$-graph consisting of two
vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who
previously determined this Tur\'an number for $r=3$. As the key ingredient of
the proof we show that the Lagrangian of intersecting $r$-graphs is maximized
by principally intersecting $r$-graphs for $r \geq 4$.
