# What is Maximization: Definition and 52 Discussions

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

Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2}$$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...
9. ### 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. ### 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. ### Projectile motion maximization

Hello Forum, I am aware that in order to maximize the time of flight, for an object being launched from a certain level and returning to the same level, the launching angle must be theta=90 degrees. To maximize range instead (same level to same level) the angle should be theta=45 degrees (no...
12. ### Can the entropy be reduced in maximization algorithms?

In maximization algorithm like that is used in artificial intelligence, the posterior probability distribution is more likely to favour one or few outcomes than the prior probability distribution. For example, in robots learning of localization, the posterior probability given certain sensor...
13. ### Optimization of windmill power with use of gears

< Mentor Note -- thread moved to HH from the technical Engineering forums, so no HH Template is shown > Okay so I'm a freshman BE student and one of our first projects is designing a windmill that can produce a voltage of 5 for 2 seconds or longer. We are having trouble find the optimal gear...
14. ### 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. ### 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. ### 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. ### 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. ### MHB Linear Programming Formulation problem faced - Maximization problem

Hi All. I am new here and I faced some issues in formulating the objective functions and constraints for the following scenario. Could any kind souls assist in giving me some advices on how I can proceed to do so? Company Y is producing two different cookies; S cookies and E cookies. The...
19. ### 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. ### 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. ### MHB Volume Maximization Problem

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. ### Inequality Maximization

1. |(x2+2x+1)/(x2+3)|≤ M. Find the value of M when |x|≤ 3. 2. |u+v|=|u|+|v| 3. I understand that you start off by distributing the absolute value symbols into the individual terms as above. Then you maximize the numerator, using 3 as the value for x. However, my professor then...
23. ### 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. ### 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. ### 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. ### Right Triangle Maximization Problem

This is a fairly standard maximization problem in calculus, but I was wondering if anybody could help me come up with a nice geometric solution. It seems like it should be possible to make an argument based on symmetry, but I haven't quite been able to work it out yet. Note, I have already...
27. ### Multi-dimensional maximization method

I'm currently working on an expanded lifting-line problem. I've got plenty of data, but I need to find the maximum oswald efficiency factor. I've got a table of data of efficiency factors for taper and twist combinations and I need to interpolate to find the maximum in 2 directions. I don't...
28. ### Inferring a vector wrt a maximization objective

Hello, I have a dataset of 5-dimensional real-valued vector X^j={x_i: i=1,2,3,4,5} and their corresponding y^j where y^j is a real-valued number and j is the no of samples. Suppose the X^j vectors are various audio feature vectors and the y^j are corresponding user ratings. Now there will be...
29. ### MHB Linear programming maximization

Who knows solve this problem of linear programming of maximization, explaining me all the steps to reach the solution. total cost to produce the products a,b,c: 100 cost to produce the product a (x)= ? quantity to produce of product a = 5 cost to produce the product b (y)= ...
30. ### Multivariable calculus: Maximization of volume of box with constraint

Homework Statement What is the maximum possible volume of a rectangular box inscribed in a hemisphere of radius R? Assume that one face of the box lies in the planar base of the hemisphere. NOTE: For this problem, we're not allowed to use Lagrange multipliers, since we technically haven't...
31. ### 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. ### 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. ### 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. ### 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. ### Profit Maximization Problem

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. ### HelpSolve Maximization Problem: 1170X1 + 1110X2

Hi Mathematicians, I recently encountered this problem in maximization... Maximize 1170X1 + 1110X2 Subject to 1. 9x1 +5x2 ≥ 500 2. 7x1 +9x2 ≥ 300 3. 5x1 + 3x2 ≤ 1500 4. 7x1 + 9x2 ≤ 1900 5. 2x1 + 4x2 ≤ 1000 X1,X2 ≥ 0 Where it was stated that 9x1 +5x2 ≥ 500 and 7x1 + 9x2 ≤...
37. ### 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. ### Interesting maximization problem

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. ### 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. ### 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. ### Maximization with constraint

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. ### Maximization of a |F|^2

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. ### Maximization of the sum of mutual information terms

I would like to compute the power distribution that maximizes the sum data rate of a certain communication scheme. The expression follows from the sum of the rate of four messages which interfere with each other. A certain amount of power (here 1 Watt) is assigned to these messages, and there is...
44. ### Profit maximization equation

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. ### Constrained Maximization

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. ### 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. ### Linear Programming and Maximization

Homework Statement AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units of...
48. ### Maximization subject equality constraint

Homework Statement Max f = x²yz³ sub. 50x+10y+100z = 1000 Homework Equations Using Lagrange: L = x²yz³ - λ ( 50x + 10 y + 100z - 1000 ) Lx = 2xyz³ - λ50 = 0 Ly = x²z³ - λ10 = 0 Lz = 3x²yz² - λ100 = 0 i found z = 2,5 x= 10 y=25, what's wrong? The Attempt at a...
49. ### Maximize Profit: Find Optimal Production with C(x) & R(x)

Homework Statement Find the level of production x that will maximize profit. Homework Equations C(x) = 500 + 100x^2 + x^3, where x = units produced. R(x) = 7000x - 80x^2 The Attempt at a Solution Should I use marginal cost and marginal revenue, or is there a way to...
50. ### Maximizing Box Volume with 1200cm^2 of Material

OK, I've been killing myself over this one problem and I just cannot seem to get it. :grumpy: I know it's probably a lot easier than I'm making it out to be. If anyone can give me a little help I would really appreciate it. Here's the question: If 1200cm^2 of material is available to make...