EXPECTED KEY RESULTS
====================

Coding Exercise 1 / mixed constraints
-------------------------------------
max x-y^2 s.t. x^2+y^2=4, x>=0, y>=0
x* = (2,0), objective = 2
Equality multiplier (slide convention) mu = 1/4.
The y>=0 constraint binds but its multiplier is zero.

Coding Exercise 2 / firm application
------------------------------------
max Pi = 90*x1 - 1.5*x1^2 + 40*x2 - 2*x2^2
s.t. x1+x2<=33, x1<=27, x2<=27
x* = (26,7), Pi=1508
Only x1+x2<=33 binds; multiplier/shadow price = 12.
A per-unit tax of 12 with the total cap removed reproduces (26,7).

Coding Exercises 3/4
--------------------
Using the equations in the supplied code:
x* ~= (3.51212134, 0.21698794, 3.55217115)
mu* ~= (1.22346356, 0.27493710)

Note: the slide's numerical hint displays x2=0.127. That value does not satisfy
8*x1+14*x2+7*x3=56. The code-consistent value is approximately 0.21699.

Equality exercise: linear objective on sphere
---------------------------------------------
max 2x-2y+z s.t. x^2+y^2+z^2=9
maximum: (2,-2,1), lambda=1/2, f=9
minimum: (-2,2,-1), lambda=-1/2, f=-9

Utility maximization
--------------------
max log(x1)+log(x2) s.t. 3x1+2x2=2
x*=(1/3,1/2), lambda=1
bordered Hessian determinant = 72

Inequality KKT exercise
-----------------------
max -(x1-4)^2-(x2-4)^2
s.t. x1+x2<=4, x1+3*x2<=9
x*=(2,2), lambda=(4,0)
Only the first inequality binds with a positive multiplier.
