Using only real numbers between 0 and 100, inclusive, show the set of three numbers whose product is 64 and whose sum is minimal is give by {4, 4, 4}.

(a) When is an absolute minimum or maximum guaranteed?

(b) State the steps to find an absolute minimum and maximum.

(c) Is the space closed and bounded? Explain.

(d) Use Lagrange Multipliers to find the minimum and maximum

please label and write neatly.

Consider an SEIR model with both horizontal and vertical transmission. What assumption can you make about the new born of mothers from the E and I compartment. Should the infected new born enter the E compartment or I compartment or both? Think of the possibilities. Draw a transfer diagram according to the different assumptions you made and derive the corresponding differential equations.
Note that :
S = susceptible
E = exposed
I = infectious
R = recovery

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is an eigenvalue of p(T), then λ is an eigenvalue of T. Under the additional assumption that V is a complex vector space, and conclude that {μ | λ an eigenvalue of p(T)} = {p(λ) | λan eigenvalue of T}.

If two operators do not commute, show that we cannot simultaneously diagonalise them .

In questions below determine whether each of the following sets is countable or uncountable.

For those that are countably infinite exhibit a one-to-one correspondence between the set of positive integers and

that set.

1) The set of positive rational numbers that can be written with denominators less than 3.

2) The set of irrational numbers between sqrt(2) and π/2.

Calculate the Frenet framing, curvature and torsion of the helix

(a cos(t), a sin(t), bt)

2. Two hundred managers from various levels were randomly selected and interviewed regarding their concern about environmental issues. The response of each person was tallied into one of the three categories: No concern, some concern and great concern. The results were: Level of Management No Concern Some concern Great concern Top Management 15 13 12 Middle Management 20 19 21 Supervisor 776 Group Leader 28 21 31 Use the .01 significance level to determine whether there is a relationship between management level and environmental concern through Chi Square

A function of two variables being continuous means

Select all that apply

The function is built from elementary functions and algebraic operations.

We can evaluate limits of the function by simply plugging in values.

Its graph can be drawn without lifting up the pencil

The function's value at each point of the domain is equal to its limit there.

All the partial derivatives exist.

Suppose h, k, r, s, t∈Z.Set a=3rt(2s+t) and b=3rs(s+2t).Prove the cubic polynomial

f (x) = (x − h)(x − a − h)(x − b − h) + k
passes through the point (h,k), has integer roots, has local extrema with integer coordinates, and has an inflection

point with integer coordinates.

How to do a value stream map (VSM) of the customer ordering process for the X-opoly scenario?

X-Opoly, Inc., was founded by two first-year college students to produce a knockoff real estate board game similar to the popular Parker Brothers; game Monopoly®. Initially, the partners started the company just to produce a board game based on popular local landmarks in their small college town, as a way to help pay for their college expenses. However, the game was a big success and because they enjoyed running their own business, they decided to pursue the business full-time after graduation.

X-Opoly has grown rapidly over the last couple of years, designing and producing custom real estate trading games for universities, municipalities, chambers of commerce, and lately even some businesses. Orders range from a couple of hundred games to an occasional order for several thousand. This year X-Opoly expects to sell 50,000 units and projects that its sales will grow 25 percent annually for the next five years.

X-Opoly’s orders are either for a new game board that has not been produced before, or repeat orders for a game that was previously produced. If the order is for a new game, the client first meets with a graphic designer from X-Opoly’s art department and the actual game board is designed. The design of the board can take anywhere from a few hours to several weeks, depending on how much the client has thought about the game before the meeting. All design work is done on personal computers.

After the client approves the design, a copy of the computer file containing the design is transferred electronically to the printing department. Workers in the printing department load the file onto their own personal computers and print out the board design on special decals, 19.25 inches by 19.25 inches, using high-quality color inkjet printers. The side of the decal that is printed on is usually light gray, and the other side contains an adhesive that is covered by a removable backing.

The printing department is also responsible for printing the property cards, game cards, and money. The money is printed on colored-paper using standard laser printers. Ten copies of a particular denomination are printed on each 8.5-inch by 11-inch piece of paper. The money is then moved to the cutting department, where it is cut into individual bills. The property cards and game cards are produced similarly, the major difference being that they are printed on material resembling posterboard.

In addition to cutting the money, game cards, and property cards, the cutting department also cuts the cardboard that serves as the substrate for the actual game board. The game board consists of two boards created by cutting a single 19-inch by 19.25-inch piece of cardboard in half, yielding two boards each measuring 19.25 inches by 9.5 inches. After being cut, game boards, money, and cards are stored in totes in a work-in-process area and delivered to the appropriate station on the assembly line as needed.

Because of its explosive growth, X-Opoly’s assembly line was never formally planned. It simply evolved into the 19 stations shown in the following table.

A polynomial in Z[x] is said to be primitive if the greatest common divisor of its coefficients is 1. Prove the product of two primitive polynomials is primitive. [Hint: Use proof by contradiction.]

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right inverse of ? . When ? has an inverse, we say ? is invertible.

Show that

(a)  ? has a left inverse iff ? is injective.

(b) If ? is a basis for ?V and ? is a basis for ?, then [?]^?_?(Transformation from basis ? to ?) has a left inverse iff its columns are linearly independent.

Use Matlab to solve the system x²+xy³=9 , 3x²y-y³ =4 using Newton's method for nonlinear system. use each of initial guesses (x₀,y₀)=(1.2,2.5), (-2,2.5), (-1.2,-2.5), (2,-2.5) observe which root to which the method converges, the number of iterates required and the speed of convergence.

Write the system in the form f(u) = 0, and report for each case the number of iterations needed for ||f(u)||²≤ 10^-12−.