A good question

[Wesley1982]

What was the Buddha's way of solving objective problems?..

[plwk]

And how is 'objective problem' defined?

[Wesley1982]

For example, persistent technical "problems" that can come from a relative or family member.

[plwk]

'technical problems' need 'technical details' if ever 'objective answers' are sought...

[Ogyen]

For example, persistent technical "problems" that can come from a relative or family member.

'Persistent technical problems' Like they refuse to work with their wheelchair and make you do it, or refuse to use a cane and are blind and keep stumbling on stuff?

In my experience, 'objective problems' come from 'subjective views', therefore, addressing the subjective view involved with kindness to help the person untie the knots they're putting forth that create the 'objective problem' thrown your way would be most beneficial. Caveat: this is super broad, since objective problems could be your brother in law beating your sister every day.. in which case, the police are needed.

[Wesley1982]

I am probably thinking of specific stuff that goes on in retirement homes and assisted living facilities.

The nurses and assistants probably know what I'm talking about.

[Wesley1982]

In my experience, 'objective problems' come from 'subjective views', therefore, addressing the subjective view involved with kindness to help the person untie the knots they're putting forth that create the 'objective problem' thrown your way would be most beneficial. Caveat: this is super broad, since objective problems could be your brother in law beating your sister every day.. in which case, the police are needed.

Police are not needed in this instance. I've seen many & many police on the news media channels.

[Wesley1982]

Back to the OP ~ how would the Buddha treat family members with chores of responsibility?..

[Andrew108]

Help them as much as possible. Just help according to your capacity to help. There is no particular philosophy involved. No need to be buddhist about it.