Which best describes the philosophes’ approach to understanding the world? they applied principles of reason. they used intuition and imagination. they engaged in research. they devised theories and conclusions

The philosophes were philosophers, we might say, within France during the period known as the Enlightenment. We might also call them social critics or social thinkers.

Let's work through the options you offer for their approach to understanding the world:
1.  They applied principles of reason. ... This sounds to me like the best way to put it.  The whole theme of the Enlightenment (which these philosophes were bringing into being) was an emphasis on using our reason to understand things better and make improvements.  "Dare to know!  Have the courage to use your intelligence!" one Enlightenment thinker said.
2.  They used intuition and imagination. ...  Many philosophes were wonderfully imaginative writers -- Voltaire, for instance.  But I'd not say that was their primary way of understanding the world.
3. They engaged in research. ...  Yes, they did.  Montesquieu's "Spirit of the Laws," for instance, gave historical observations about governments and what works where.  But again, overall, while there were methods of research the philosophes' biggest emphasis was on the use of reason to think things through.
4.  They devised theories and conclusions. ... Again, yes they did.  But wouldn't "devising theories and conclusions" be another way of saying they applied principles of reason?

I'd go with choice #1.

Slander.