You appeared to have copied the problem wrong. . No matter what t is, x is never less than -1 and, in particular, is never equal to -3. Since the "boudaries" given do not intersect, they cannot bound a region.
Mathematics requires great precision. If you do not take the time and care to read the problem correctly you are are never going to get anywhere. " " has exactly the same difficulty as " ". so that x is never less than -1. The curve given by , does not intersect x= -3 so the curves you give do NOT bound a region.
If it were , when or , , t= -1 and t= 3. At those points, y= 1 and 9.
Now, find the gradient of F(x,y) and set it equal to 0 to find any points in the interior of the region at which there may be max or min.
Then set x=-3 in F(x,y) to get a function in the single variable y giving the value of F on the line x= -3. You can set the derivative of that equal to 0 to find any critical points there (remember that y must be between 1 and 9 to be on that boundary).
Set , in F to get a function of the single variable t on the parabola. Set its derivative equal to 0 to find any critical points there (remember that t must be between -1 and 3).
Evaluate the function F(x,y) at each of the critical points you have found, as well as at the vertices (-3, 1) and (-3, 9) to determine the largest and smallest values of F in that region.