Total cost (TC) and total variable cost (TVC) are parallel, yet average total cost (ATC) and average

Question:

Total cost (TC) and total variable cost (TVC) are parallel, yet average total cost (ATC) and average variable cost (AVC) are not parallel.
a. Demonstrate mathematically that ATC and AVC are not parallel.
b. Show mathematically that when both ATC and AVC are falling, ATC falls faster than AVC, and when both are rising, AVC rises faster than ATC.