You do not need to unmount the root partition to extend it with this method.
You may or may not need to shut down the VM for the qm resize. I haven't yet tested without shutting down first, I'll try it next time and update this document accordingly.
First in proxmox resize the image:¶
qm resize <vmid> <disk> <amount>
qm resize 116 virtio0 +32G
Next launch vm and run¶
Resize the partition in the partition table with fdisk or parted, I use fdisk:
fdisk /dev/vda delete partition 3 (or whichever is the LVM partition) recreated partition 3 at the same start sector, use the new end sector
DO NOT ERASE THE LVM INFORMATION!
Partition #3 contains a LVM2_member signature. Do you want to remove the signature? [Y]es/[N]o: N
Resize the LVM partition you edited with parted or fdisk
Next extend it to use the free space:
lvextend -l +100%FREE /dev/ubuntu-vg/ubuntu-lv