Swansea's relegation was then confirmed over the weekend following a 2-1 defeat at home to Stoke.
Swansea were bottom of the Premier League and five points adrift of safety when Carvalhal was appointed at the end of December following his departure from Sheffield Wednesday a few days earlier.
Carlos Carvalhal insists Swansea must fight for their "little hope" in trying to remain a Premier League club.
"We reached the quarter-final of the FA Cup as you know, which was a good achievement, and in the Premier League we have had 17 games". There have certainly been mistakes along the way and it is down to the owners to shoulder much of the responsibility.
"I don't believe too many managers would have done better than we did".
"But we are taking steps to rectify that and we will be bringing additional resources to that department during the summer in a bid to change and strengthen the way we operate".
"In many ways we need to rediscover that identity both on the pitch and off it if we are to make the community proud of this football club again". We believe this is a time for everybody to pull together even through the heartbreak and bitter disappointment of relegation. The Swans have picked up three points and scored only two goals over their last eight games.
It's not a question of firing - Carvalhal defiant on Swansea future
Paul Clement turned the ship around and delivered a seventh successive top-flight campaign but was replaced by Carlos Carvalhal in December a year ago with the club bottom.
Gayle helped Newcastle back into the top-flight at the first time of asking last season, notching 23 times as Rafa Benitez's men won the Championship title. Before we arrived, there were 13 losses, with us we lost seven.
Fans groups are so disenchanted they have called on long-standing chairman Huw Jenkins to resign - blaming him for his part in the 2016 takeover and a botched recruitment policy.
Carvalhal got off to a flying start, only for results to tail off, with the club's seven year stay officially coming to an end on Sunday following a loss to Paul Lambert's Stoke, who will also be in the second-tier next season.
"To achieve one play-off is very hard and to repeat, it is so difficult that usually teams do not repeat it".
Promoted under manager Brendan Rodgers in 2011, Swansea finished 11th in their first season back in the top flight.
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