Assignment:
Q1. 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)
Q2. 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.
Provide complete and step by step solution for the question and show calculations and use formulas.