people need energy to live./ to live with.

1、people need energy to live.
不定式“to live”做的是状语,此时可以把to改成"in order to",因为live可以做不及物动词,所以全句很完整.不需要加with.
2、I need a pen to write with.
不定式“to write with”做的是定语,修饰a pen,此时write作不及物动词,并与该名词构成动宾关系,即“write with a pen”,所以需要带相应的介词.
3、此外,write也可作及物动词,如：write a composition (作文).
“我有一篇作文要写”就可以写成：I have a composition to write.
根据2、3句可以知道,不定式作定语时带不带介词,关键要看不定式与其所修饰的名词之间需不需用到介词.