Consider the following function:
f (x , y , z ) = x 2 + y 2 + z 2 − x y − y z + x + z
(a) This function has one critical point. Find it.
(b) Compute the Hessian of f , and use it to determine whether
the critical point is a local man, local min, or neither?
(c) Is the critical point a global max, global min, or neither?
Justify your answer.
Consider a function f(x) which satisfies the following
properties:
1. f(x+y)=f(x) * f(y)
2. f(0) does not equal to 0
3. f'(0)=1
Then:
a) Show that f(0)=1. (Hint: use the fact that 0+0=0)
b) Show that f(x) does not equal to 0 for all x. (Hint: use y=
-x with conditions (1) and (2) above.)
c) Use the definition of the derivative to show that f'(x)=f(x)
for all real numbers x
d) let g(x) satisfy properties (1)-(3) above and let...
Consider the following production function: f(x,y)=x+y^0.5. If
the input prices of x and y are wx and wy respectively, then find
out the combination of x and y that minimizes cost in order to
produce output level q. Also find the cost function.
Find the maximum and minimum values, and at what point they
occur.
A. F(x,y)= 25-16x-2x2+8y+4y2 over the
region bounded by x=5, y= -2, and y=x
B. F(x,y)= x2y+3xy-4y+15x over the region bound by
x=0, x=3, y= -4, and y=4
Consider the following.
optimize f(t, w) = 5t − t2 + 2w − w2
subject to g(t, w) = 2t + w = 14
(a) Write the Lagrange system of partial derivative equations.
(Enter your answers as a comma-separated list of equations. Use λ
to represent the Lagrange multiplier.)
(b) Locate the optimal point of the constrained system.
(t, w, f(t, w)) =
(c) Identify the optimal point as either a maximum p
Consider the following.
optimize f(r,
p) = 3r2 +
rp − p2 +
p
subject
to g(r, p) =
3r + 4p = 1
(a) Write the Lagrange system of partial derivative equations.
(Enter your answer as a comma-separated list of equations. Use
λ to represent the Lagrange multiplier.)
(b) Locate the optimal point of the constrained system. (Enter
an exact number as an integer, fraction, or decimal.)
Once you have the answer matrix on the homescreen of your
calculator,...
On R2, consider the function f(x, y) = ( .5y,
.5sinx). Show that f is a strict contraction on R2. Is
the Banach contraction principle applicable here? If so, how many
fixed points are there? Can you guess the fixed point?
Consider the following linear programming problem
Maximize
$4X1 + $5X2
Subject To
2X1 + 5X2 ≤ 40 hr
Constraint A
3X1 + 3X2 ≤ 30 hr
Constraint B
X1, X2 ≥ 0
Constraint C
if A and B are the two binding constraints.
(Round to ONLY two digits after decimal
points)
a) What is the range of optimality of
the objective function?
Answer ≤ C1/C2 ≤ Answer
b) Suppose that the unit revenues for X1 and X2 are changed to
$100 and...