Maximization Definition and 52 Threads

Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and Josh Stewart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my utility?" It is a type of optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending (income), the prices of the goods and their preferences.

Utility maximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are rational, they seek to extract the most benefit for themselves. However, due to bounded rationality and other biases, consumers sometimes pick bundles that do not necessarily maximize their utility. The utility maximization bundle of the consumer is also not set and can change over time depending on their individual preferences of goods, price changes and increases or decreases in income.

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  1. L

    Lagrange multipliers understanding

    Here’s my basic understanding of Lagrange multiplier problems: A typical Lagrange multiplier problem might be to maximise f(x,y)=x^2-y^2 with the constraint that x^2+y^2=1 which is a circle of radius 1 that lie on the x-y plane. The points on the circle are the points (x,y) that satisfy the...
  2. LCSphysicist

    Question about reflectivity and maximization of energy reflected

    Imagine that we have a transmitter of microwaves that radiates a linearly polarized wave whose E field is known to be parallel to the dipole direction. We wish to reflect as much energy as possible off the surface of a pond (having an index of refraction of 9.0). Find the necessary incident...
  3. T

    Maximization of a Multivariable Function

    The beginning is straight forward and I found f=x^2-2yz, which satisfies grad(f)=F. Then I calculated W= f(x,y,z)-f(0,1,1) since it's conservative. I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to...
  4. opus

    Maximize Fencing for 4 Equal Pastures - 125,000 Linear Feet

    Homework Statement A rancher has 125,000 linear feet of fencing and wants to enclose a rectangular field and then divide it into four equal pastures with three internal fences parallel to one of the rectangular sides. What are the dimensions for each of the four equal pastures that will...
  5. F

    I Expectation Maximization (EM) : find all parameters

    I am tackling a technique to determine the parameters of a Moffat Point Spread Function (PSF) defined by: ## \text {PSF} (r, c) = \bigg (1 + \dfrac {r ^ 2 + c ^ 2} {\alpha ^ 2} \bigg) ^ {- \beta} ## with the parameter "(r, c) =" line, column "(not necessarily integers). The observation of a...
  6. K

    Maximization problem — Stiffest beam that can be cut from a log

    Homework Statement Homework Equations Pitagora's: ##~a^2+b^2=c^2## Maxima/minima are where the first derivative is 0 The Attempt at a Solution $$\left( \frac{a}{2} \right)^2+\left( \frac{b}{2} \right)^2=r^2~\rightarrow~b^2=\frac{16r^2}{a^2}$$ The strength S has the proportion coefficient k...
  7. petterson

    A Maximization problem using Euler Lagrange

    Hi, I'm trying to solve the following problem ##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##. I have only little experience with calculus of variations - the problem resembles something like ## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt## but I don't know about the...
  8. J

    A Maximization Problem: Double Int. w/ C not Dependent on Integrals

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  9. S

    Maximization problem with simplex method

    Homework Statement Maximize the profit function P = 3x - y - 2z subject to the following x - y - z ≤ 15 2x - y + 2z ≤ 50 2x + y + z ≤ 39 , where x ≥ 0, y ≥ 0, z ≥ 0. Homework Equations Simplex method / Simplex algorithm The Attempt at a Solution Hello to everyone who is reading this. :)...
  10. rkum99

    Maximization of motor speed/torque

    For an upcoming competition, I must construct a vehicle that is capable of traveling at high speeds and can travel precise predetermined distances. While any electrical components may be used, any and all circuitry can only be powered by a single 9V battery. I am currently using a rather...
  11. F

    Projectile motion maximization

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  12. A

    Can the entropy be reduced in maximization algorithms?

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  13. T

    Optimization of windmill power with use of gears

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  14. N

    Distance formula maximization problem

    Homework Statement At 9 P.M. an oil tanker traveling west in the ocean at 18 kilometers per hour passes the same spot as a luxury liner that arrived at the same spot at 8 P.M. while traveling north at 23 kilometers per hour. If the "spot" is represented by the origin, find the location of the...
  15. F

    Why this maximization approach fails?

    Homework Statement Find all points at which the direction of fastest change of the function f(x,y) = x^2 + y^2 -2x - 2y is in the direction of <1,1>. Homework Equations <\nabla f = \frac{\delta f}{\delta x} , \frac{\delta f}{\delta y} , \frac{\delta f}{\delta z}> The Attempt at a Solution...
  16. K

    Maximizing Angular Acceleration

    Homework Statement A massless stick of length d, held parallel to the ground, has a mass, m, attached at one end of it and a pivot on the other end. A second mass, m, is glued on at a distance x from the pivot. At what distance x would maximize the angular acceleration of the stick the instant...
  17. T

    Maximization of an Uncertainty Product

    Homework Statement [/B] Sakurai problem 1.20: find the linear combination of spin-up and spin-down S_z eigenkets that maximizes the uncertainty product \langle(\Delta S_x)^2\rangle\langle(\Delta S_y)^2\rangle. Homework Equations [/B] In general, we can write a normalized spin-space ket as...
  18. H

    MHB Linear Programming Formulation problem faced - Maximization problem

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  19. B

    MHB Lagrangian utility maximization with a ''complex'' summation

    Hello there! It's my first time posting here, I hope you guys will be good to me :). I took a one year break to study a language abroad, and now it seems like I forgot everything math-wise. I'm preparing for a test and I'm having a really hard time doing the following problem. I need to...
  20. V

    Maximization of wave interference

    I was thinking about which two musical notes, when played together, maximize the absolute value of the area under the curve of the resulting wave within its period while keeping the amplitude to a minimum (ideally the amplitude of the original two sound waves). I guess you could try to maximize...
  21. L

    MHB Thank you for your understanding.

    A tree trunk is shaped like a truncated cone it has 2 m of length and diameters of their bases are 10 cm and 20 cm. Cut a square straight section so that the axis of the beam coincides with the axis of the truncated cone. find the beam volume maximum that can be drawn from this form. answer...
  22. A

    What is the value of M when |x|≤ 3 in the inequality (x^2+2x+1)/(x^2+3) ≤ M?

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  23. MarkFL

    MHB Collin's question at Yahoo Answers regarding maximization of beam strength

    I have posted a link there to this thread so the OP can view my work.
  24. MarkFL

    MHB Maximizing Volume of a Cuboid - Nthabiseng P's Q&A

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  25. MarkFL

    MHB S N's question at Yahoo Answers regarding revenue maximization

    Here is the question: I have posted a link to this topic so the OP can see my work.
  26. X

    Right Triangle Maximization Problem

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  27. I

    Multi-dimensional maximization method

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  28. A

    Inferring a vector wrt a maximization objective

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  29. E

    MHB Linear programming maximization

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  30. M

    Multivariable calculus: Maximization of volume of box with constraint

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  31. G

    Simple maximization question very confused :s

    Hey everybody! My question is: Find the value of x that maximizes the following function and the maximum value (a is constant): f(x) = x^2 subject to 0 ≤ x ≤ a. It is supposed to be solved without calculus and I'm terrible confused! how would i go about solving this? wen i plot the curve...
  32. S

    Question about probabilistic combinatorial maximization

    Hello, I have some question about probabilistic combinatorial maximization as follows: Let X = {X_1, ..., X_n} be a set of i.i.d. positive random variables, S = {s_i} be a set of all combinations of selecting m r.v.'s from X, and Y(s_i) = the sum of r.v.'s in the combination s_i ...
  33. B

    Maximization problem of tuple set

    Given a set of vectors {v_ j } = {v_1, ... v_N} and I wish to transform each vector in the following manner v_ i ' = Sum_k=1 to N ( c_ k,i) * v_i where c_ k,i is a scalar and what we are trying to solve for. such that the sum of the distances squared between each pair of transformed vectors...
  34. M

    Algorithm to solve a maximization problem

    I have two sets of vectors: A: a1, a2, a3... an B: b1, b2, b3... bm n > m and n/m is an integer, p. Each vector bi has ranked, in order of preference, a set of vectors from A. For example, b1 may "prefer" a1, a9, and a10. The only constraint on this set is that each vector ai from A appears...
  35. L

    How to Maximize Profit for Book Publisher?

    1. A publisher wants to dispose books. For 400 copies or less the price is $30 per book. For orders of more than 400 the price of each book is dropped by 2 cents for each extra book ordered beyond 400. The cost of production is $6 for each book. Find a formula for the profit function and how...
  36. chwala

    HelpSolve Maximization Problem: 1170X1 + 1110X2

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  37. Z

    Maximize Daily Profits with Cost and Price Functions | Homework Help

    Homework Statement daily Cost function C(x) = 5x + 360 -0.001x^2, where x is the number of decks company produces each day and daily cost is in dollars. Suppose that the price that each deck is sold for varies based on the equation given by p(x) = 11.30 - 0.01x, where p is the price per deck...
  38. F

    How to Maximize Your Chances of Stopping a Truck in the Desert?

    Homework Statement Imagine that you are lost in the desert. You are equipped with a compass. You can use your compass to measure angles, but not the distances. There is a road (going from east to west) and you're going northwards. Suddenly, you spot a truck (on a road, going to west) at...
  39. I

    Utility maximization of potential disasters

    I've been doing some thinking on what resources should be allocated to stymie anthropogenic global warming. Some figures: Anywhere between 88-98% of publishing climate scientists believe in some form of AGW, based on over 6 large scale surveys. In one survey, 41% of meteorologists and...
  40. C

    General question about solving force maximization problems

    I'm doing this E&M problem for fun (I'll be taking the class this fall), but now it's just starting to get frustrating. I can't really post the problem because there is a diagram, but I'm not really looking to be given the actual solution anyway, so I'll describe it to you: Two positive...
  41. H

    How Do You Maximize a Quadratic Form on a Sphere with an Affine Transformation?

    Hi, I search for the maximum of a quadric for points on a sphere. I have an affine transform A (4x4 matrix, in homogeneous coord.) and apply it to points on (and inside) a sphere x \in S_{m,r} \Leftrightarrow (x-m)^2<=r^2 . (Although I think the extremum must be on the surface of the sphere?)...
  42. N

    Maximization of |F|^2 with Constraints on Real Inputs

    Will anyone please help me to solve the problem: F(x_1,x_2,...,x_n) is a complex valued function and each x_i are real (may be positive too) numbers. I have to find the maximum of |F| (or |F|^2) w.r.t. x_i. What are the set of constraints? I don't think it will be exactly as...
  43. M

    Maximization of the sum of mutual information terms

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  44. R

    Maximizing Profits: How Many Shares to Buy for the Best Return in 6 Months?

    Hey guys I just came across this website i think its awsome..! Hope i can contribute tooo! I had a question: If a Share costs 3 dollars, and it doubles every 3 months And nother share costs 1 dollar and doubles in One month How many shares of each would you buy with 1000...
  45. M

    What Is the Correct Objective Function for Maximizing Package Volume?

    Homework Statement A particular parcel sercive will accept only packages with length no more than 128 inches and length plus girth(xz)(width times height) no more than 145 inches. What are the dimensions of the largest volume package the parcel service will accept? This is the problem...
  46. N

    WriterMaximization of Differentiable Real-Valued Function with Linear Variables

    Hi everyone, Suppose f =f(x_1, x_2,...,x_n) be a real-valued, any-time differentiable function. Let each x_i=x_i(u_1, u_2,...,u_{2^n-1}) be a linear function of reall u_i's. Let f=g(u_1, u_2,...,u_{2^n-1}). Then is it right that Max f w.r.t. x_i=Max of g w.r.t. u_i? Sorry for the...
  47. R

    Linear Programming and Maximization

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  48. O

    Maximization subject equality constraint

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  49. A

    Maximize Profit: Find Optimal Production with C(x) & R(x)

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  50. Apost8

    Maximizing Box Volume with 1200cm^2 of Material

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