Interesting word that... how does it make you feel? Probably starting with a definition might help. I'm going to ignore, at least for the moment, the 'universal' definition or understanding about this word and look to scripture for a description, definition. Starting with Google first though - this is what 'they' say about the word.
"In scripture, the word "examine" (often linked with "test" or "prove") is frequently used to encourage self-reflection and spiritual assessment. It's an invitation to introspectively evaluate one's faith, actions, and motivations in light of God's word and teachings."
Purpose?
"This examination is not meant to induce self-condemnation, but rather to foster growth, identify areas needing improvement, and ensure one's faith is genuine."
Before commenting and getting into a discussion, let's also look at what the Word says:
" But a person must [prayerfully] examine himself [and his relationship to Christ], and only when he has done so should he eat of the bread and drink of the cup." [1 Corinthians 11:28]
Most people are familiar with this description as it is part of the commentary when the church observes Holy Communion.
"Test and evaluate yourselves to see whether you are in the faith and living your lives as [committed] believers. Examine yourselves [not me]! Or do you not recognize this about yourselves [by an ongoing experience] that Jesus Christ is in you—unless indeed you fail the test and are rejected as counterfeit?"
[2 Corinthians 13:5]
I think the question with this verse is: do we? Have you defined, not just described, 'committed believers'? Back to the question of how have you defined 'examine'? Let me also suggest Psalm 51:10-19 and 26.
When you get right down to it - examine is an 'ouchy' word. It can be very painful when you do an honest appraisal of yourself. When you do take the time and examine yourself: your motives, the continuity between your words and your actions, the good bits as well as the sinful bits. All need considering. I never said it was easy. And yet, it is one of the most freeing and directing activities we do.
Well I've set the parameters (self), the context (examine) and now to the work of examining. This was the introduction - stay around for the next segment.