The Shrew got Taimed?

The protagonist of the play The Taming of the Shrew by William Shakespeare is Katherine. At the beginning when Petruchio, a gentleman of Verona, was courting Katherine, the headstrong and stubborn shrew, he was faced with a challenge. Katherine is at first an unwilling to accept their relationship but Petruchio tempers her with various psychological torments. She also realizes that if she does not marry she will either end up as a nun or an old, lonely woman. The "taming" takes place while the couple is about to marry and after the couple marries.
At the beginning of the play this character showed that she was strong, foul-tempered and sharp-tongued. She insults and degrades the men around her. For example, in Scene 1-Act 1 she retaliates towards Hortensio by saying: “I’ faith, sir, you shall never need to fear. Iwis it is not halfway to her heart. But if it were, doubt not her care should be/ To comb your noodle with a three-legged stool/ And paint your face and use you like a fool. She insults him because he states earlier that she should be gentler and milder.
In Act 2-Scene 1, Katherine and Petruchio have a fast paced conversation in which they go back and forth play fighting almost. She acts like a shrew to him and he responds the same way. Petruchio does not mind it actually seems like he likes it. That continues throughout the play until Act 4-Scene 5 when there is a point where she seems kind of changed. She agrees more with Petruchio and is as though she just does not want to oppose anything he says. The behavior she reveals suggests that Petruchio might be forcing her or taming her. He does not feed her and he does not let her sleep. At the end of the story, she is considered to be fully tamed. Although she acts as if she is fully tamed through her articulate speech in Act 5-Scene 2, I believe that she might have just put a face for the evening. Or maybe Shakespeare had to end it this way because if he did not what was going to be the resolution of his story. The title of the play is TheTaming of the Shrew and if the Shrew was not tamed, what was the point of the story? I do not think that it would have been the same ending if it would have been written in this time. Divorce was unheard of and if that was an option Katherine would have been old and lonely. That was probably the only alternative she had. Or maybe she just acted that way in public but at their home it was a different story?