3
votes
2answers
212 views

What is a French curve, as mentioned by Feynman?

I'm reading "Surely You're Joking, Mr. Feynman!", he says: I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a ...
-4
votes
0answers
16 views

Optimisation: Maximum of a rectangle with semi circles at each end

A field is being built in the form of a rectangle with semi circles at each end. A $400$m racetract to is be built around the playing field. a) What Radius of the semicircular end would give the ...
0
votes
0answers
20 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
-1
votes
3answers
30 views

Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
0
votes
1answer
13 views

Determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3 metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and front (BC) are ...
-1
votes
1answer
28 views

determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3  metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and  front (BC) are ...
4
votes
1answer
106 views

find the minimum value of $x^2-6x+9+ \dfrac{64}{x^2}$

Looking for an elegant solution. I can do by brute force, that is finding derivative and double derivative. All Ideas will be appreciated and tried by me.
-1
votes
1answer
20 views

Nearest and farthest point from a function to another [closed]

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
-1
votes
1answer
36 views

By looking at a graph of a function, how do I find the maxima/minima of its curvature function. [closed]

All Ideas are appreciated. I could think of some intuitive ideas, but could not back them by solid clean reasoning. thanks, I will post If i find anything
5
votes
4answers
143 views

How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$

Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$ My idea: I guess $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le ...
1
vote
1answer
43 views

revenue optimization under multinomial logit

Let $[N]= \{1,...,N\}$ denote a set of items, item $i$ has an utility equal to $u_i > 0$ and a unit revenue of $r_i >0 $. Without loss of generality, assume that $$r_1 \geq r_2 \geq ... \geq ...
1
vote
1answer
76 views

Optimize $x^2 + y^2 +2z^2 +z(x^2-y^2)$ subject to $x+y=2$

$$x^{2}+y^{2}+2z^{2}+zx^{2}-zy^{2}\overset{\left(x=2-y\right)}{\longrightarrow}4-4y+2y^{2}+2z^{2}+4z-4yz\rightarrow FOC: \; \begin{cases} -4+4y-4z=0\\ 4z+4-4y=0 \end{cases}\rightarrow y=1+z\rightarrow ...
2
votes
1answer
19 views

Norman Window Optimization

A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 24 ft that will admit the greatest possible amount of light. I know that I ...
0
votes
1answer
40 views

Optimization Calculus Problem- Flight

If exactly 230 people sign up for a charter flight, the operators of a charter airline charge Dollars 330 for a round-trip ticket. However, if more than 230 people sign up for the flight, then fare is ...
0
votes
2answers
41 views

Rectangular Box Optimization Problem

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs \$0.31 per square foot, the material for the sides costs $0.05 per square foot, and the material ...
1
vote
2answers
56 views

Minimizing Question

A closed box constructed from a tin sheet has a square base and a volume of $343 \text{in}^3$. Find the dimensions of the box, assuming the minimum amount of material was used in its construction. ...
0
votes
2answers
24 views

Optimize volume of an open cardboard box made from flat square of cardboard…

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. ...
0
votes
2answers
22 views

A simple optimization problem of reciprocal function

Can someone tell me the answer to this question? I cannot seem to figure it out The function $y=\frac{2}{x}$ is decreasing in?? a.$(0,\infty)$ b.$(-\infty,0)$ c.$(0,2)$ d,$(-\infty,\infty)$ I ...
0
votes
1answer
17 views

geometric significance of the largest possible dimensions of a rectangle

Find the largest possible rectangular area you can enclose, assuming you have 128 meters of fencing. what is the (geometric) significance of the dimensions of this largest possible enclosure? My ...
1
vote
0answers
35 views

optimization word problem in calculus

You are asked to build an open cylindrical can (i.e. no top) that will hold $665.5$ cubic inches. To do this, you will cut its bottom from a square of metal and form its curved side by bending a ...
1
vote
1answer
36 views

Optimization question for calculus

Could anyone tell me where is my mistake? I took the derivative and I solved for r and ended up with this answer
1
vote
3answers
30 views

Optimization in Calculus

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.
0
votes
0answers
30 views

Minimizing a multivariable function in several variables

I would like to show that a certain function is negative, to help establish asymptotic stability via a Lyapunov function for a system of differential equations. This is exactly what I need help on: ...
1
vote
2answers
74 views

When are there no critical points?

Is there ever a time when there are no critical points of a function? For example, I am trying to find the critical points and the extrema of $\displaystyle f(x)= \frac{x}{x-3}$ in $[4,7]$ I am not ...
1
vote
1answer
31 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
0
votes
0answers
16 views

How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
1
vote
0answers
26 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
1
vote
1answer
58 views

Half Sphere Optimization

Having a little trouble with an optimization question: ...
1
vote
4answers
83 views

Local minimum of $f(x) = 4x + \frac{9\pi^2}{x} + \sin x$

What's the minimum value of the function $$f(x) = 4x + \frac{9\pi^2}{x} + \sin x$$ for $0 < x < +\infty$? The answer should be $12\pi - 1$, but I get stuck with the expression involving both ...
-1
votes
0answers
49 views

Find the angle between hypotenuse and the side.

For a right angled triangle, the sum of the length of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle. Find the angle between hypotenuse and the side.
1
vote
3answers
57 views

Extrema of $f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$.

(a) I need help in finding maxima and minima of the following funcion: $$f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$$ therefore I need to find the roots of ...
0
votes
2answers
59 views

Optimization problem, not sure how to proceed

So I'm a bit confused by this optimization word problem. I would be able to solve it I think given number values for the speeds but I'm uncertain how to get an exact answer when you don't know the ...
1
vote
2answers
82 views

How to find what are the points closest to and farthest from (0,0) of ellipse $9x^2+4y^2=36$ using optimization?

Please do not use Lagrange multipliers. Assume these have not been introduced and optimize. Edit: I try optimizing the squared distance formula using the equation as a constraint, but I only get one ...
0
votes
0answers
12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
1
vote
1answer
39 views

extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
1
vote
3answers
42 views

Calculate minimum perimeter of a rectangle with an extra constraint.

I have been set this problem, and although I can derive a minimum perimeter using calculus, I now need to add an extra constraint to one side of the rectangle and I am having problems deriving a ...
0
votes
1answer
60 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
0
votes
1answer
26 views

Calculus optimization quick question

A hotel fills only $120$ rooms when the price is $\$150$ per night for a room. When the price is decreased by $\$10$, it fills $16$ additional rooms. Find the price for maximum revenue. Ok frankly ...
1
vote
1answer
28 views

Maximization with constraints

How can I find $\lambda_H$ and $\lambda_T$ such that $$\max_{0 \leq \lambda_H , \ \lambda_T \ \leq 1 }\left\{\frac{4.6575342 \times 10^{-4}}{2.1722965 \times 10^{-4} + \lambda_H},\frac{1.0958904 ...
2
votes
1answer
16 views

Minimization of two variables

(General) How can I find $$\min\limits_{\phi\leq x\leq \alpha , \ \beta \leq y \leq \delta}\{a+bx,c+dy\}$$ given values of $a,b,c,d,\phi, \alpha, \beta, \delta \in \mathbb{R}?$ (Specific) How can I ...
1
vote
3answers
290 views

How to find the minimum/maximum distance of a point from elipse

I have the point $(1,-1)$ and the ellipse $$x^2/9 + y^2/5 = 1 $$ How to find the minimum and maximum distance of the point from the ellipse ? from exploring the ellipse I know that $$a = 3$$ , $$b ...
1
vote
0answers
26 views

Local extrema given the graph of a function's derivative

I am given a graph of the derivative of a function and answered most of the questions, but am still stuck at answering where the local extrema are. I had a sample question to reference from and it ...
0
votes
1answer
73 views

Weighted Singular Value Decomposition

Lemma: $\forall A\in R^{n\times n}$ and a diagonal matrix $\forall W\in R^{n\times n}$ with $ w_{11}\geq w_{22}\geq ...\geq w_{_{nn}} >0$. The singular value decomposition of A denoted by: $A=XM ...
0
votes
2answers
33 views

How to find this maximum

We have $$n\in\mathbb{N}\quad k=1,...,n$$ we want to find $$\max_k{\cos(\frac{k\pi}{n+1})}$$ As we don't have a continuous application , we have a set of $n$ points we cannot do the typical ...
0
votes
4answers
58 views

The sum of two variable positive numbers is $200$. Find the maximum value of their product.

The sum of two variable positive numbers is $200$. Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$. NB: I'm currently on the ...
0
votes
0answers
10 views

Learning a multivariate polynomial with dependent coefficients

I have a polynomial of the form of $ K^2((a-i)^2 + (b-j)^2 + c^2) = (ct)^2$ where $a,b,c,t$ are unknowns. I have multiple observation points for the values of $i,j,K$. Can I use some technique to ...
0
votes
1answer
16 views

Finding coefficients of a third degree polynomial

The third degree polynomial $$-x^3 + ax^2+bx+c$$ has an maximum at $(2,10)$ and an inflation point at $(0,-6)$. Find the coefficients $a$ $b$ and $c$. Am I supposed to differentiate the polynomial ...
0
votes
2answers
46 views

fence a circular land and a square land.

With a wire mesh of 1000 mts divided into two parts , we want to fence a circular land and a square land. a)Calculate the lengths of each of the parties such that the total area enclosed is ...
0
votes
1answer
28 views

Find f with A plane curve whose equation is $y - f (x) = 0$ passes through the origin.

A plane curve whose equation is $y - f (x) = 0$ passes through the origin.Consider the rectangle $R_x$ formed by the coordinate axes and lines parallel to the axis passing through the point $(x, f ...
1
vote
2answers
30 views

Does Lagrange multiplier have solution if functions doesn't intersect

I am trying to get intuition behind Lagrange multiplier and question that bothers me is: Does Lagrange multiplier have solution if two functions(main function and constraint) doesn't intersect. Thank ...