Functions of several variables

Constrained Optimization Problems

1. Lagrange Multiplier Problems

1. Consider the function $f(x,y) = e^x(2x^2-xy+y^2)$ a) Find all critical points. b) Classify these critical points.

2. For the function $f(x,y) = x^2+3y$, analyze under the constraint $\frac{x^2}{4} + \frac{y^2}{9} = 1$ a) Use Lagrange multipliers to find critical points. b) Determine the maximum and minimum values of the function on the constraint.

3. Consider the function $f(x, y) = x^2+y^2-3xy+x$ a) Determine the gradient. b) Classify all the critical points. c) Determine the maximum and the minimum of the function $f$ on the constraint $D = {(x, y) : x^2+y^2 = \frac{1}{3}}$.

2. Unconstrained Critical Point Analysis

1. Determine the domain of the following functions; compute their gradient; find all the critical points and classify the critical points different from $O = (0, 0)$: a) $f(x, y) := x(y+1)^2 - x^2y$ b) $f(x, y) := x^3 + y^2 - x^2y$