Question

Find the extreme values of f on the region described by the inequality. f(x, y) = 2x2 + 3y2 − 4x − 2, x2 + y2 ≤ 16

Answer 1

Please, use "^" to denote exponentiation.  Your 2x2 should be written as 2x^2.

Your relationships are then f(x,y)=2x^2+3y^2-4x-2 
and                                                 x^2 + y^2 ≤ 16

Please note that the 2nd relationship represents a circle of radius 4 whose entire interior has been shaded.

Graph this.  Then, rewrite f(x,y) in the form f(x,y) = 2x^2 - 4x  + 3y^2 -2.  Can you identify the shape of the curve in the xy plane representing this function?  You'll need to find the points (x,y) in which the graph of f(x,y) intersects the shaded circle x^2 + y^2 ≤ 16. 

Think:  what do "extreme values of f"