Corner Roof Fascia Damage
Looking at the side view of the fascia board, there's
a good chance that more of the fascia board is damaged, the roofing
material doesn't over hang the gutter enough and therefore water is
allowed to accumulate in between the fascia board and the gutter.
Over time this can cause the moisture to seep into the wood, creating wood rot (like the picture above) and sometimes attracting termites.
You can also see that the plywood nailed underneath the roof rafters is also falling apart and it seems like the most damage is in the corner area. The barge rafter, the fascia board and the plywood used to cover the eaves seems to have the most damage in the corner.
While doing any home repairs, just don't fix the damage, you need to find out where the damage is coming from. If you repair the damage and don't fix the problem, eventually you're going to have to fix the same problem over again.
Home Repair Tip: Whenever your installing gutters on your home, make sure that the roofing overlaps the gutters enough to actually go into the gutter. On lower sloped roofs, this could be a bigger problem than steeper ones.
For Example: the water won't be running off the roof as fast on a low sloped roof and could actually roll back underneath the roof shingle and onto the wood framing located underneath the edge of the roof.
A solution to this problem, would be to install a piece of thin galvanized or aluminum metal underneath the roof shingle and building paper and let the piece of metal protrude at least 1 inch past the edge of the roof shingles.
The purpose for the metal would be to extend the length of the roof shingles, more towards the center of the gutter.