On R2, consider the function f(x, y) = ( .5y,
.5sinx). Show that f is a...
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 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.
6.14 Let f = {(x, y) ∈
R2 : y = x5 + 4x3 + x +
1}.
Prove that (a) f is onto. (b) f is 1-1.
Prove that g = {(x, y) ∈ R2 : x =
y5 + 4y3 + y + 1} is a function. (You will
need to use calculus to prove part (1).)
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...
f(x,y)=30(1-y)^2*x*e^(-x/y). x>0. 0<y<1.
a). show that f(y) the marginal density function of Y is a Beta
random variable with parameters alfa=3 and Beta=3.
b). show that f(x|y) the conditional density function of X given
Y=y is a Gamma random variable with parameters alfa=2 and
beta=y.
c). set up how would you find P(1<X<3|Y=.5). you do not
have to do any calculations
1. In the utility function U(x,y)=x^5y^2
does Y exhibit diminishing marginal rate of substitution? Please
show your work.
2.What is the no-waste condition for the utility function
U(x,y)=min{x/3,y/5}
express your answer in terms of y as a function of x
3. What is the MRS of the utility function U(x,y)=3x+1/2y?
please show your work
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.
f(x,y) = 2/7(2x + 5y) for 0 < x < 1, 0 < y < 1
given X is the number of students who get an A on test 1
given Y is the number of students who get an A on test 2
find the probability that more then 90% students got an A test 2
given that 85 % got an A on test 1
Please Consider the function f : R -> R given by f(x, y) = (2
- y, 2 - x).
(a) Prove that f is an isometry.
(b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C =
(3, 2), and the triangle with vertices f(A), f(B), f(C).
(c) Is f a rotation, a translation, or a glide reflection?
Explain your answer.
Consider the function f(x, y) = 4xy − 2x 4 − y
2 .
(a) Find the critical points of f.
(b) Use the second partials test to classify the critical
points.
(c) Show that f does not have a global minimum.