In your opinion, what view of youth and old age does Beowulf convey? In answering, consider not only the details in the last part of the poem but also the earlier portrayals of Beowulf and Hrothgar.

Neither youth nor age is demonized in Beowulf. Both are presented as periods with their strengths and weaknesses.
The youthful Beowulf is energetic and proud. He wants to show off his physical prowess and become a hero. He boasts a little, but his actions show he is as strong as he claims to be.
During the first two-thirds of the poem, King Hrothgar is the representative of old age. He acts as a father figure to the young Beowulf. He is not physically strong and so cannot protect his people against monsters on his own as Beowulf does, but he is wiser, advising Beowulf against ignoring the higher spiritual treasures of heaven and taking all the credit for his abilities for himself without giving thanks to God.
In the final third of the poem, Beowulf has become an old man, yet he is not robbed of his dignity. He is not as powerful as he once was and even dies in battle, but he is wiser and more responsible, now acting as a king to his people. He also becomes a father figure to a young warrior named Wiglaf. This echo of the Beowulf/Hrothgar dynamic in the earlier parts of the story shows how youth and old age are parts of a cycle, neither necessarily better than the other, but natural parts of life.

At the beginning of Beowulf, Hrothgar is portrayed as a generous and benevolent king who does a great deal for his people, such as building them the splendid mead-hall, Heorot, in which to feast. However, although he was a fearsome warrior in his youth, his age means that he cannot offer the one thing the Danes need above all else: protection from Grendel. For this, he has to call upon the young hero, Beowulf.
Fifty years later, Beowulf does not make the same mistake. When he can no longer protect his people, he dies in combat, passing on the crown to Wiglaf, the only one of the Geats who distinguished himself in the battle with the dragon and managed to survive it.
The poem depicts a primitive society in which a leader needs to be able to defend himself and his people. Once he is too old to do this, whatever his other qualities, he has outlived his usefulness in a perpetually dangerous world where youth, strength, and heroism are in constant demand.

In Beowulf, youth and age are not treated very differently from the way we treat them in the contemporary West. Youth is a time of learning, characterized by seeking adventure and having to prove oneself through accomplishments. Old age is associated with wisdom, and leadership goes to those who have proven themselves through time.
This can be seen clearly by comparing the characters of Beowulf and Hrothgar at the beginning of the poem. Hrothgar is a well-respected and loved leader who has ruled the Danes effectively and successfully. His age, and the experience that comes with it, have brought power and honor to him.
A few other details hint at the respect given to the old. The poem begins with a call out to the ancient kings, describing some of Hrothgar's ancestral line, which gives honor to those old kings. Beowulf, in another demonstration of the social status of the elderly, is told by the "old men" or "wise men" (depending on the translation) that he should go across the sea to help the Danes. The old are the source of wisdom and knowledge.
Beowulf is a much younger man than Hrothgar, and as a younger man his actions are more geared toward building and maintaining a reputation as a great warrior. Beowulf listens to the advice of the elderly wise of his people, and he chooses acts of bravery that will prove his strength and courage.
Finally, when Beowulf has fought his final battle at the end of the epic, the concept of youth as a proving ground is reiterated when he hands the throne over to Wiglaf, the only Geat who stood by him while he fought the dragon. Wiglaf has proven himself, and Beowulf, now a wise old man, has chosen the most worthy man to be his successor.