13:4 For he was crucified through weakness,
yet he lives through the power of God.
For we also are weak in him,
but we will live with him through the power of God toward you.

13:11 Finally, brothers, rejoice.
Be perfected, be comforted, be of the same mind, live in peace,
and the God of love and peace will be with you.
13:12 Greet one another with a holy kiss.
13:13 All the saints greet you.
13:14 The grace of the Lord Jesus Christ, the love of God,
and the communion of the Holy Spirit, be with you all. Amen.

Paul has to deal with two kind of apostates. In 1Corinthians he dealt with moral apostates - the licentious Christians. He warns that he may deal with these again when he comes if they haven't changed their behavior. The other apostates are the doctrinal apostates, most likely the group of the circumcision with its legalistic philosophy. Later John deals with another form of doctrinal apostates - the gnostics. Today there are many deviations from Biblical Christianity. But if we are not allowed to speak of such differences and to judge right from wrong, then such leaven may propagate unhindered in the Christian community.
 

When Paul commands them to examine themselves, he was not speaking on an individual level alone. Yes, he wanted people to examine themselves individually, but also the context indicates that he wanted them to examine others in the church, as well as examining himself, as verse 6 indicates, "And I trust that you will discover that we have not failed the test." (niv)

But how do we go about examining ourselves and others as to whether such are in the faith or not? The book of 1st John goes into this extensively. But Paul had already mentioned some principles. For he had already told them, for example:

"Do you not know that the wicked will not inherit the kingdom of God? Do not be deceived: Neither the sexually immoral nor idolaters nor adulterers nor male prostitutes nor homosexual offenders nor thieves nor the greedy nor drunkards nor slanderers nor swindlers will inherit the kingdom of God." 1Cor 6:9,10 (niv)

But understand also that this process may not be as clearcut as it seems. For in the case of the moral apostates, one must judge relative to the person's overall lifestyle as a Christian. For even those born of God will sin from time to time. Thus when John writes, "No one who is born of God will continue to sin, because God's seed remains in him; he cannot go on sinning, because he has been born of God." (1John 3:9 niv) He is speaking in a lifestyle sense. Also with doctrinal apostates it may not be so easy. For there are a variety of theologies in the Christian community, as I point out on the site, which deviate to different degrees from Biblical truth. But to what extent such ideas deviate from the truth before they can be reckoned as indicating heretical apostacy may not be clear cut.

John reckoned the docetists (those who considered Christ as coming in spirit only, and not in physical flesh) as being apostates and never having been born of God (1John 2:19) Paul reckons the group of the circumcision who teach that one must be circumcised and follow the law of Moses to be saved, also to be apostates - false brothers (Gal 2:4) But what about, for example, those who believe in a purgatory, like the Catholics? Or what about those who believe that salvation can be lost, like the Arminians? What about those who believe that a person can be saved and continue to live a lifestyle of sin without any affect on his salvation status? It's easier to identify apostates for the more overt sins and for the more heretical ideas. The greater these deviate or conform to Biblical Chrisianity is the degree to which we can be confident in discerning correctly as to whether such people are in the faith or not.

vs 7-9 I think what Paul means is that even if they were to misjudge Paul, he still desires that at least they would behave properly according to the truth. He is not referring to misjudging his teachings, for they are the truth. But rather misjudging his motivation and attitudes. So Paul is saying that even if you despise me as a person, I will continue to exhort and pray for you to walk according to the truth. Which is quite different than the world's attitude of: "You don't like me? Well then to hell with you!"

Praying for "perfection" is praying for their maturity in the faith. A maturity which will give them the discernment necessary to recognize truth from error, right behavior from wrong behavior, and true believers from false.

vs 10 Just as in the case of parental authority, there are times when one must use it reluctantly in a harsh manner in order to instill a sense of discipline. The absence of such discipline in the midst of unrepentant rebellious licentiousness only helps to promote such behavior further. If we don't control ourselves, then God's authorities will help us to do so. And if we reject them, then God will use even more harsh authorities, like Satan.  For hadn't Paul handed over one of the Corinthians Christians in 1Cor 5:5 to Satan? Not a pleasant thought. But as he pointed out in 2Cor chapter 2, such harshness led them to repentance.

vs 11 The first two commands are in the passive voice (in greek) - be perfected, be instructed (by what I write).
The second two commands are in the active voice - be of one mind, live in peace.
Being perfected or matured is much a function of how we respond to what comes into our lives. And similarly with instruction. These come from outside and we need to respond to them properly. But to be of one mind and live in peace takes active initiative.

There are many commands to be united and be of one mind. But much of the Biblical concept of unity is misunderstood in much of modern Christianity. The Biblical concept of unity is that we should all be united around Biblical truth. But unBiblical unity is centered around whatever is popular. The problem is that whatever is popular is seldom Biblical. Thus those who hold to Biblical truth are often viewed as being divisive, since they unite around a different center. And as I mentioned before, the philosophy of pluralism and "feel-good" Christianity develops only a surface unity in the Christian community, as differences are not allowed to be discussed. Thus Biblical truth and applications of Biblical truth are eliminated from the do-nothing Christian community for the sake of such "unity." Living in peace should be modeled after Christ and the Apostles, who while they lived in peace with true believers, often experienced conflict with the elite religious community.

vs 14 Here's a picture of the trinity. Grace was provided by Christ, an effect of God's love resulting in an intimate relationship with God experienced through the indwelling of the Holy Spirit.
The KJV also includes the note:
<<The second epistle to the Corinthians was written from Philippi, a city of Macedonia, by Titus and Lucas.>> (Meaning that Paul dictated the letter to them.)