Is it musthib to read auzubillah silently in namaz?

You can, I mean I don't know if it's musthib or not but you can. Some sheikhs, when coming to read say it out loud others say it to themselves in a whisper.

Sana-Praise to Almighty Allah and Tauwaz are recited in a whisper and Sura Fatiha is read aloud by the Imam. While offering prayer alone, whole Namaz is offered in whisper. However, if you are at some lonely place, you may recite Fatiha aloud in Fajr, Maghrib and Isha prayers.