Consider the function on the interval (0, 2π). f(x) = sin(x)/ 2 + (cos(x))^2 (a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing (b) Apply the First Derivative Test to identify the relative extrema. relative maximum (x, y) = relative minimum (x, y) =

Find the dimensions (in inches) of the rectangular package of maximum volume subject to the constraint that the sum of the length and the girth cannot exceed 192 inches (see figure). (Hint: Maximize V = xyz subject to the constraint x + 2y + 2z = 192.)

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring (with addition and multiplication of matrices defined as usual).

-3,7 11,-15

A. Find Point Slope Form

B. Find Slope Intercept

C. Find X and Y intercepts

Sketch a graph

Show that of all isosceles triangles with a given perimeter , the one with greatest area is equilateral.

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x, y) = x³y + 24x² − 8y

The demand equation for your company's virtual reality video headsets is

p=1000/q^.3

where q is the total number of headsets that your company can sell in a week at a price of p dollars. The total manufacturing and shipping cost amounts to $140 per headset.
(a) What is the greatest profit your company can make in a week? (Give your answer to the nearest whole number.)

How many headsets will your company sell at this level of profit? (Give your answer to the nearest whole number.)

(b) How much, to the nearest $1, should your company charge per headset for the maximum profit?

For the given polynomial, find all zeros of the polynomial algebraically. Factor the polynomial completely. ?(?) = ?^4 − 2?^3 − 2?^2 − 2? − 3

For the given polynomial, find all zeros of the polynomial algebraically. Factor the polynomial completely. ?(?) = 6?^4 − 7?^3 − 12?^2 + 3? + 2

Let f(x) = 1 + x − x² +e^x-1.

(a) Find the second Taylor polynomial T₂(x) for f(x) based at b = 1.

b) Find (and justify) an error bound for |f(x) − T2(x)| on the interval
[0.9, 1.1]. The f(x) - T₂(x) is absolute value.

Please answer both questions cause it will be hard to post them separately.

An investor is considering three types of investments: a high risk venture into oil leases with a potential return of 15%, a medium risk investment in bonds with a 9% return, and a relatively safe stock investment with a 5% return. He has $50,000 to invest. Because of the risk, he will limit his investments in oil leases and bonds to 30% and his investments in oil leases and stock to 50%. How much should he invest in each to maximize his return, assuming investment returns are as expected?

a. Define the variables

b. Clearly state the constraints (all inequalities) related to the feasible region

c. State the objective function

d. Set up the initial simplex matrix needed to solve the linear programming problem using the Simplex Method

e. Perform all pivots necessary using row operations to transform the matrix until the solution is feasible

f. How much should he invest in each to maximize his return, assuming investment returns are as expected?

prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) - f(x)g'(x))/g^2(x)

A cylindrical can is to be built to occupy a volume of 5000m3. The top and bottom of the cylinder cost $500/m2, wheras the cost to build the wall of the cylinder is $300/m2. Also, the top of the cylinder is to have a circular hole that will occupy 1/4 of the top area. Find the dimensions, to 2 decimal places, of the cylinder to minimize the cost. What is the cost, rounded to the nearest hundred, to build the cylinder?

use the Rational Zeros Theorem to find all the real zeros of each polynomial function given below: f(X)=3X^3 + 6X^2 -15X - 30

f(X) = 2X^4 - X^3 - 5X^2 + 2X + 2

** Using Calculus**

We have a client asking for our recommendations on which storage containers are better for shipping and storing hazardous waste. Their requirements are as follows: The container should hold 50,000 cm3 (which is 50 liters); The sides of the container will be made from aluminum sheets with a thickness of 2 mm; The two bases of the container need to be thicker (for greater support) and will be made from aluminum sheets with a thickness of 6 mm; There are two options to consider for the shape of the containers: Option 1: closed cylinders; Option 2: closed boxes with a square base (note the top and bottom are squares, but the sides could be rectangles). Some considerations to take into account: The fixed cost of production of a cylinder container is $10 per cylinder; The fixed cost of production of a box container is $4 per box; The current price for 2mm aluminum sheets is $0.011 per square centimeter. The current price for 6mm aluminum sheets is $0.045 per square centimeter; You need to advise the client on the dimensions of the containers that will both meet the size requirement and minimize cost. Based on the cost, you should make a recommendation to the client on whether square or cylinder containers should be used.

Given a 2x2 matrix, A =

1 4

2 -1

Find its eigen values, eigen vectors. Can matrix be diagnolized?