All uses of take it easy I can think of are nearly interchangeable with relax. (One wouldn't say just "relax" as a farewell, but in that use it's similar to "stay cool" or "have a good [relaxed] day".)
And uses where take it easy is literal imperative advice are not necessarily reproachful.
"Augh, I'll never get all these facts straight in time for the test! I'm going to fail, I know it!" "Whoa, take it easy. You can do this."
Do you remember Click and Clack, the Tappet Brothers, hosts of the off-beat radio show called Car Talk? Their website has a list of classic (and very funny) "Famous Last Words", including these:
- He's probably just hibernating.
- What does this button do?
- Are you sure the power is off?
- Let it down slowly.
- These are the good kind of mushrooms.
- It's strong enough for both of us.
- The odds of that happening have to be a million to one!
- I'll hold it and you light the fuse.
- Don't be so superstitious.
If you understand the spirit of "famous last words," each one of these should elicit a chuckle. Obvious, #1 is talking about a bear – a bear that wasn't just hibernating. As for #2, I don't know what that button does, but I don't think the speaker should have pressed it! In #3, they thought they had turned the power off, but I guess they shut off the wrong circuit breaker. I suppose in #4, the speaker is standing under something heavy, like a piano – just before the rope snaps. And on it goes.
The phrase "famous last words" is more than just skeptical or dubious, there's a hint of irony mixed in, with maybe even a bit of superstition, bad karma, or Murphy's Law, too. Going back to the example you cited:
We won't miss the train. Mike's never late.
Can't you just feel how such an utterance increases the probability that Mike's going to be late, and we're going to miss the train? Instead of saying, "Famous last words," I could reply by saying this – it would convey about the same meaning:
Don't say that – you'll jinx us.
or:
There's a first time for everything!
The "last words" in the expression "famous last words" could be the last words ever spoken, if the utterance preceded some fatal calamity. However, they could just as easily refer to the last words spoken "before the accident happened," or "before we called the ambulance," "before we missed the train," or, "before the house caught on fire."
Here are a few more to think about:
I do this all the time!
Don't worry, I'm sure the ice is thick enough.
I'm positive this answer won't get downvoted.
Famous last words.
Best Answer
What you are thinking of is make it up to somebody. That means to do something good for someone you have upset to be on good terms with them again.