Carrots are loaded with beta-carotene, a powerful antioxidant, and beet juice has been known to reduce inflammation.
Shred carrots, celery root, radish, apple, and beet in the food processor or on the coarse grate of a box grater, ending with the beets so that they don't stain the other vegetables. Toss together in a large bowl and set aside.
Whisk together the lemon juice and vinegar. Whisk in yogurt and Dijon. Gradually drizzle in the olive oil, whisking constantly to make an emulsified dressing. Season with salt and pepper. Add dressing and herbs to shredded root vegetables and toss with a spatula. Serve cold or at room temperature.