My mother had a future protection mandate drafted by her notary and appointed me as the agent. Unfortunately, she passed away before the mandate was activated. I am her sole heir. Can I use this mandate to retrieve funds (15,000 euros) ...

... from her bank accounts? 

No. The future protection mandate ceases to apply upon the death of the person to be protected (article 483 2° of the Civil Code). You should consult a notary to establish a certificate of heirship that will allow you to recover your mother's liquid assets (article L 312-1-4 of the Monetary and Financial Code).