There is five people in my family including me , me and my wife and three children which are less than age of 10. and i am the only the person who earns money to run my family . should i have to make qurbani separately by name of each person of my family . our its permissible to make qurbani behalf of Mohammed Sallallahu alai wa sallam.

Sahih Muslim Hadith 4845     Narrated by Aisha

Allah's Messenger (saws) commanded that a ram with black legs, black belly and black (circles) round the eyes should be brought to him, so that he should sacrifice it.  He (saws) said to Aisha: “Give me the large knife”, and then said: “Sharpen it on a stone.” She did that. He (saws) then took it (the knife) and then the ram; he placed it on the ground and then sacrificed it, saying: “In the name of Allah, O Allah, accept (this sacrifice) on behalf of Muhammad, and the family of Muhammad, and the Ummah of Muhammad".

 

Sunan of Abu-Dawood Hadith 2789        Narrated by Jabir ibn Abdullah

The Prophet (saws) sacrificed two horned rams which were white with black markings and had been castrated. When he made them face the qiblah, he said: “I have turned my face towards Him Who created the heavens and the earth, following Abraham's religion, the true in faith, and I am not one of the polytheists.  My prayer, and my service of sacrifice, my life and my death are all for Allah, the Lord of the Universe, Who has no partner. That is what I was commanded to do, and I am one of the Muslims. O Allah it comes from Thee and is given to Thee from Muhammad and his Ummah. In the name of Allah, and Allah is Most Great. He then made sacrifice.”

It is Sunnah to declare who is offering the sacrifice.  If one wishes and possesses the means, one may sacrifice an animal for each member of their family, or one may sacrifice only one sheep for oneself and one's whole family or household.