(x^3-3x^2-x+3)/(x^2+x-x) find critical numbers ,relative min/max and graph the function

A farmer has 150 acres of land suitable for cultivating Crops A and B. The cost of cultivating Crop A is $40/acre, and the cost of cultivating Crop B is $60/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of Crop A requires 20 labor-hours, and each acre of Crop B requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. He has also decided that he will cultivate at least 80 acres of Crop A. If he expects to make a profit of $180/acre on Crop A and $200/acre on Crop B, how many acres of each crop should he plant to maximize his profit?

What is the maximum profit?
$

Based on Fibonacci's sequence, prove that the equation x^3 + 2x^2 + 10x = 20 can have no solution in the form a + rad(b), where a and b are positive rationals.

Consider the relation T on the set of all undergraduate TAs (UTA) for the CSE department where s₁ T s₂ if and only if s₁ and s₂ are UTAs for the same course.

(a) Assuming that no one is a UTA for multiple courses, prove that T is an equivalence relation.

(b) Assuming that no one is a UTA for multiple courses, what do the equivalence classes for T represent?

(c) Explain why the assumption that no one is a UTA for multiple courses is necessary for T to be an equivalence relation.

1. (a) Explain what is meant by the terms countable set and uncountable set. Give a concrete example of each.

b) Show that if A and B are countably infinite then the set A x B is also countably infinite.

c) Give an expression for the cardinality of set A U B when A and B are both finite sets.

d) What can you say about the cardinality of the set A U B when A and B are infinite sets.

Determine whether the polynomial f(x) is irreducible over the indicated field. (a) f(x) = 4x^ 3 + 10x^2 + 16x − 11 over Q (b) f(x) = 7x^ 4 − 9x^ 2 + 6 over Q (c) f(x) = x^3 − x + 1 over Q (d) f(x) = x ^4 + 2x ^2 + 2 over Q (e) f(x) = x^ 4 + 2x^ 2 + 2 over C (f) f(x) = x ^4 + 2x^ 2 + 2 over R (g) f(x) = x ^2 + 2x − 3 over Z-5 (h) f(x) = x ^3 − 2 over Z-7

Polynomial and rational functions can be used to model a wide variety of phenomena of science, technology, and everyday life.

Choose one of these sectors and give an example of a polynomial or rational function modeling a situation in that sector. [Hint: see the examples and exercises in the book.]

Go to www.desmos.com/calculator, write your equation, or function, and develop your explanation using the properties of graphs.

Your Discussion should be a minimum of 250 words in length and not more than 750 words.

Dear expert, I will submit this assignment after an hour, please don't copy from internet and add APA text citation if needed.

Let F be a field and let φ : F → F be a ring isomorphism. Define Fix φ to be Fix φ = {a ∈ F | φ(a) = a}. That is, Fix φ is the set of all elements of F that are fixed under φ. Prove that Fix φ is a field.   (b) Define φ : C → C by φ(a + bi) = a − bi. Take for granted that φ is a ring isomorphism (we proved this in class at some point). Find Fix φ.

Use the method of reduction of order to find a second solution y2 of the given differential equation such that {y1, y2} is a fundamental set of solutions on the given interval.

t²y′′ +2ty′ −2y=0, t > 0, y1(t)=t

(a) Verify that the two solutions that you have obtained are linearly independent.

(b) Let y(1) = y0, y′(1) = v0. Solve the initial value problem. What is the longest interval on which the initial value problem is certain to have a unique twice differentiable solution?

(c) Now suppose that y(0) = y0, y′(0) = v0. Is there a solution in this case (how does it depend on y0 and v0? If so, how many? Explain.

Let R and S be commutative rings with unity. (a) Let I be an ideal of R and let J be an ideal of S. Prove that I × J = {(a, b) | a ∈ I, b ∈ J} is an ideal of R × S. (b) (Harder!) Let L be any ideal of R × S. Prove that there exists an ideal I of R and an ideal J of S such that L = I × J.

Fill in the missing algebraic steps in the concrete renormalization calculation for period doubling.Let f(x) = -(1 + mu)x+x^2. Expand p+(n)subn+1 =

f^2(p+(n)subnetting) in power of small deviations (n) subn using the fact that p is a fixed point of f^2

Q1. Define/outline the following:

a. the difference between a ‘one-tailed test’ and a ‘two-tailed test’.

b. the importance of sample size in the context of OLS regression.

*c. four different types of data structures and discuss their potential usefulness and application in finance.

d. the OLS assumptions

e. Correlation is not causation. Briefly discuss.

(c. is the part I'm confused most about)

5. Find the generating function for the number of ways to create a bunch of n balloons selected from white, gold, and blue balloons so that the bunch contains at least one white balloon, at least one gold balloon, and at most two blue balloons. How many ways are there to create a bunch of 10 balloons subject to these requirements?

You wish to retire in 18 years , at which time , you want to have accumulated enough money to receive an annuity of $500 monthly for 20 years of retirement. During the period before retirement you can earn 4% annually,while after retirement you can earn 6 percent on your money. What monthly contributions to the retirement fund will allow you to receive the 500 dollars annuity?

X and Y are subsets of a universal set U.

Is the following statement true or false? Support your answer with a venn diagram.

X - (Y u Z) = (X - Y) n (A - C)