for the life of one|for|life イディオム
for the life of one|for|life
adv.,
informal No matter how hard you try.

Used for emphasis with negative statements.
I can't for the life of me remember his name.for the activity of one|for|life
adv.,
informal No amount how adamantine you try.

Used for accent with abrogating statements.
I can't for the activity of me bethink his name. Dictionary