1. Consider the function below. By straightforward inspection (no differentiation is necessary), determine the coordinates (x,y) of the point of minimum. Further, by using the standard procedure involving differentiation, determine the coordinates (x,y) of all critical points (you are not required to classify which point is minimum or maximum or saddle). Is the above mentioned point of minimum among them?
f(x,y) = (x^2 + y^2)e^(y^2-x^2)
2. Determine which of the following expressions represent the exact differential and find the corresponding function satisfying the condition f(-1,2) = 5, if the function exists.
(a) (5y^2 - 12x^2y)dx + (10xy - 4x^3)dy;
(b) ye^(2y)dx + xe^(2y)dy.
This solution involves determining critical points and calculating points regarding expressions. The problems involve determining the maximum points of a function using straightforward inspection, and determining expressions that represent the exact differential and corresponding functions satisfying a certain condition.