In Genesis 37:14, Jacob commands his son Joseph:
|יד וַיֹּאמֶר לוֹ, לֶךְ-נָא רְאֵה אֶת-שְׁלוֹם אַחֶיךָ וְאֶת-שְׁלוֹם הַצֹּאן, וַהֲשִׁבֵנִי, דָּבָר; וַיִּשְׁלָחֵהוּ מֵעֵמֶק חֶבְרוֹן, וַיָּבֹא שְׁכֶמָה.||14 And he said to him: 'Go now, see whether it is well with thy brethren, and well with the flock; and bring me back word.' So he sent him out of the vale of Hebron, and he came to Shechem.|
Joseph goes to seek out his brothers. He finds them, and they are upset when he approaches. They decide to conspire against him to kill him and call out "Behold, the dreamer is coming." Due to this episode, Joseph is eventually sold, after which he works for Potiphar, is thrown in jail, and finally interprets the Pharoah's dreams, which leads him to becoming extremely powerful.
Similarly, in I Samuel 17:17-18, Jesse commands his son David:
The JPS translation here does not give you the exact parallel, but this verse can be interpreted so that in fact Jesse was asking his son David to check on his brothers' welfare.
When David comes to the camp, his eldest brother upbraids him for his presumptuousness, telling him he is 'evil of heart' in verse 28. Due to this mission, David is in the right place at the right time when it comes to the fighting of Goliath, his fame spreads throughout the land, and eventually he becomes king.
In both cases, the youngest son (Benjamin was not yet born) goes out to seek his brothers' welfare at his father's behest. In both cases, there are brothers who do not respond positively. And in both cases, this is the beginning of the individual's ascent to power.
There are obviously more connections between David and Joseph. One of the most obvious is that each one is tempted by a beautiful woman; Joseph does not succumb but David does.
In addition, there is the traditional idea that there are two Messiahs. One is the Messiah son of Joseph, and the second is the Messiah son of David.
I have to explore the connections more to see if I can come to any conclusion as to why they are there. For now, I just want to note that they exist.