Let be a function that is bounded by
I am to find the absolute max/min values.
Completely lost! This is significantly more difficult that any examples we went through in class.
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.