Show that in any subtree of a max-heap, the root of the subtree contains the largest value occuring anywhere in the subtree.

This follows from the max-heap property.

If the $i$th element is the root of the subtree, then both its children would be less or equal to it. Since this properties holds for their children too and it is transitive, all of the descendents will be less or equal to the root, making it the largest value.