0
votes
2answers
38 views

Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
2
votes
1answer
80 views

Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
0
votes
0answers
20 views

Alternative to Hungarian Algorithm to determine minimum cost?

Is there a graphic calculator (CAS technology) method to solve minimum cost problems/allocations that are normally completed with the Hungarian Algorithm... Hungarian Algorithm is time consuming, ...
0
votes
2answers
43 views

Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
0
votes
4answers
45 views

Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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.
2
votes
1answer
43 views

Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
0
votes
2answers
57 views

Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
0
votes
1answer
77 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
3
votes
2answers
60 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
0
votes
1answer
30 views

Transform unconstrained optimization problems into constrained ones?

I want to formally show that the following minimization problem $$ \min_\theta||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ is equivalent to $$ \min_{\beta, \{w_i \}^{n}_{i=1}} ...
0
votes
0answers
20 views

Analytic solution to a maximization problem - Solve for $R$

I'm trying to use a CARA utility function $U(x)=e^{-\theta x}$ in the context of the Schumpeterian growth model to solve for the R&D spending. I set up a maximization problem \begin{equation} ...
0
votes
4answers
116 views

How to find the minimum value of $x^2+y^2+xy-4$ where $x+y=2$. [closed]

How to find the minimum value of the expression: $x^2+y^2+xy-4$ where $x+y=2$
0
votes
1answer
62 views

How do I properly set up this optimization equation?

So I've been the given the task to fully optimize any packaging. I chose a DS game box. So first I took the measurements of the cartridge itself ($3.5 \text{ cm} \times 3.3 \text{ cm} \times 0.38 ...
1
vote
1answer
322 views

Find the maximum or minimum value of the quadratic function by completing the square.

Find the maximum or minimum function of the quadratic function by completing the squares. State the value of $x$ at which the function is maximum or minimum. $y=3x^2+7x+9$ I already posted similar ...
3
votes
2answers
2k views

Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
4
votes
0answers
30 views

Is there a name for systems of equations with min and max functions included?

In a big project I'm working on, I'm running into systems of equations that look like the following: $$a = \min(b, c)$$ $$b = d^2 + a$$ $$c = \max(a + b, d)$$ Basically, nonlinear systems of ...
0
votes
0answers
24 views

Show that z is only positive when $\min(\frac{x^{0.5}}{y^{0.5}}, \frac{y^{0.5}}{x^{0.5}}) > A$?

where $$z = \frac{x - Ay^{0.5}x^{0.5}}{x + y - 2Ay^{0.5}x^{0.5}}$$ where $-1 < A < 1$. So the two conditions must be: $y > Ay^{0.5}x^{0.5}$ and $x+y > 2Ay^{0.5}x^{0.5}$ OR $y < ...
1
vote
1answer
50 views

Maxima/minima of $f(x)=\frac{\sin(\frac{1}{2} Nx) }{\sin(\frac{1}{2} x)}.$

How do I find: the $\bf maxima$ and minima of the function $f$ with $ f$ given by: $$f(x)=\frac{\sin(\frac{1}{2} Nx) }{\sin(\frac{1}{2} x)}, \;\;(N=1,2,3...)$$ What I did, is: Minima: I set: ...
1
vote
4answers
49 views

Derivatives question help

The question is :Find the derivative of $f(x)=e^c + c^x$. Assume that c is a constant. Wouldn't $f'(x)= ce^{c-1} + xc^{x-1}$. It keeps saying this answer is incorrect, What am i doing wrong?
5
votes
3answers
304 views

Minimizing the length of wire between two poles?

There are two poles (lets say poles A and B) $50$ feet apart and the poles are $15$ and $30$ feet tall. There is a wire which runs from the top of pole A to the ground, and then to the top of pole B ...
0
votes
1answer
90 views

Calculus question with optimization homework

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area ...
1
vote
3answers
122 views

How to find the point on a parabola where x and y are equal?

On a parabola how could i find the point at which the y and x points are equal and meet on a point of the graph, algebraically?
0
votes
1answer
25 views

What is the value of $[c,d]$ when $c$ and $d$ be such that $f(x) ∈ [c, d]$ for all $x ∈ [a, b]$?

Let $c$ and $d$ be such that $f(x) \in [c, d]$ for all $x \in [a, b]$. What is the value of $[c,d]$ for the function $f(x)=\sqrt{1-x^2}$ on the interval $[a, b]=[0,1]$? I knew taking the minimum and ...
0
votes
1answer
44 views

Conditions for a system to be solvable.

I have the following system of equations: $$\begin{aligned} \left\{\begin{array}{l} a+dz+cy+exy = 0\\ 10a+3bx-exy =0\\ -5a-dz = 0 \end{array}\right. \end{aligned}~~.$$ I would like to solve for ...
0
votes
1answer
37 views

$\sup_{x>0}\sqrt{\frac{2}{\pi}}\exp(x-\frac{x^2}{2})=?$

$$\sup_{x>0}\sqrt{\frac{2}{\pi}}\exp(x-\frac{x^2}{2})=?$$ I tried in the following way: $$\sup_{x>0}\sqrt{\frac{2}{\pi}}\exp(x-\frac{x^2}{2})$$ ...
0
votes
0answers
43 views

How $k^∗$ be infinite and $\mu= \frac{\lambda}{m}$?

$$k^∗ = \sup_{x>0}\frac{\lambda^m x^{(m−1)}e^{(μ−λ)x}}{μΓ(m)}$$ (1) How will $k^∗$ be infinite if $m < 1$ or $λ ≤ μ$ ? taking the derivative of the right-hand side above and set it to zero, ...
4
votes
1answer
155 views

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
2
votes
2answers
101 views

maximum using completing the square

Is it just me, or this problem does sound weird? The Parks Department is fencing a rectangular dog-run (a place for dogs to exercise) in a local park. It will be 7 yards longer than 5 times its ...
6
votes
1answer
273 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
0
votes
1answer
126 views

Optimizing $x^2+y^2$ from two given equations? [duplicate]

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to: $$2x^2+5xy+3y^2=2$$ and $$6x^2+8xy+4y^2=3$$ Note: Calculus is not allowed. I tried everything I could but whenever I got for ...
1
vote
1answer
31 views

Finding a concave function that minimize the middle value while the boundary values are fixed

This question came to me while I was listening to Dominik's talk this afternoon. First, let me remind you what does f is concave mean. It means f satisfies $pf(x)+(1-p)f(y)\le f(px+(1-p)y)$, $\forall ...
4
votes
4answers
178 views

Not so easy optimization of variables?

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to $2x^2+5xy+3y^2=2$ and $6x^2+8xy+4y^2=3$. (calculus is not allowed). I tried everything I could but whenever I got for example ...
2
votes
0answers
54 views

How to solve this type of equation with posynomial form?

I have an equation with the following form where the goal is to find $x$: $$ \sum_k c_k x^{\gamma_k} = 1$$ where $c_k, \gamma_k \in \Re^+$ and $\gamma_k > 1$ Alternatively using $y = \log(x)$ I can ...
7
votes
4answers
459 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
0
votes
1answer
81 views

How to maximize this function of X,Y?

I have 2 input $X$ and $Y$ which are both positive integers. I have to maximize this function Let $A=\min(Y/4,X/2)$ , $B=\min(W/2,Y/2)$, $C=\max(A,B)$, and $D=\max(X-W,Y)$. Then $$ ...
0
votes
3answers
123 views

How to write “the parameter maximizing the maximum of the maximum value of two functions continuous in the domain of maximization”

Say you have $f(x),g(x)$ continuous where they need to be and you want to express the following: Give me the biggest value of $f$ for $x \leq X_f$ , give me the biggest value of $g$ for $x \leq X_g$, ...
0
votes
1answer
109 views

Finding the minimum of $N = \frac{(a+3c)}{(a+2b+c)}+\frac{(7a+6b+3c)}{(a+b+2c)}+\frac{(c-a)}{(2a+b+c)}$ if $a, b, c \in \Bbb R$

Find the minimum of $$N = \frac{(a+3c)}{(a+2b+c)}+\frac{(7a+6b+3c)}{(a+b+2c)}+\frac{(c-a)}{(2a+b+c)}. \qquad (a,b,c \in \Bbb R^+)$$
3
votes
1answer
43 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
0
votes
2answers
56 views

Fit screen resolution given ratio and total number of pixels

Given: width: 1920 height: 1080 total pixels: width * height = 2073600 aspect ratio: 1920 / 1080 ~= 1.8 How do I calculate a new resolution (width and height) ...
3
votes
3answers
170 views

Finding the minimum of $\frac pq + \frac rs$ for distinct integers $p, q, r, s$ from $\{1,2,3,4,5,\ldots,16,17\}$

Here is the question: Four distinct integers $p$, $q$, $r$ and $s$ are chosen from the set $\{1, 2, 3, 4, 5, \ldots, 16, 17\}$. The minimum possible value of $\frac pq + \frac rs$ can be written ...
1
vote
1answer
87 views

Maximum/minimum problem of integers

Let $f$ be the function such that $$f(x,y,z,w)=x+w, \quad x,y,z,w\in{\Bbb Z}$$ where $$ x+y+z+w=400, $$ and $x<y<z<w$. How can I find the maximum of $f$? I think the key point is to use ...
1
vote
1answer
119 views

For integers $a$ and $b \gt 0$, and $n^2$ a sum of two square integers, does this strategy find the largest integer $x | x^2 \lt n^2(a^2 + b^2)$?

Here is some background information on the problem I am trying to solve. I start with the following equation: $n^2(a^2 + b^2) = x^2 + y^2$, where $n, a, b, x, y \in \mathbb Z$, and $a \ge b \gt 0$, ...
0
votes
0answers
82 views

Maximization of a sum subject to constraints on 3 resources

This is a generalization of a subproblem from a past programming competition that I had trouble with. Given input $6$ positive integers: $$r_1, r_2, r_3, x_1, x_2, x_3 \in \mathbb{Z^+}$$ ...
5
votes
2answers
137 views

Minimize sum of smallest and largest among integers on the real line.

Suppose there are 3 non-negative integers $x$, $y$ and $z$ on the real line. We are told that $x + y + z = 300$. Without loss of generality, assume $x$ to be the smallest integer, and $z$ to be the ...
5
votes
3answers
380 views

The minimum value of $(\frac{1}{x}-1)(\frac{1}{y}-1)(\frac{1}{z}-1)$ if $x+y+z=1$

$x, y, z$ are three distinct positive reals such that $x+y+z=1$, then the minimum possible value of $(\frac{1}{x}-1) (\frac{1}{y}-1) (\frac{1}{z}-1)$ is ? The options are: $1,4,8$ or $16$ ...
38
votes
4answers
954 views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
1
vote
2answers
198 views

How to find the minimum value of $px+qy$ when $xy=r^2$?

The question says: "Find the minimum value of $px+qy$ when $xy=r^2$." No information is given on $p,q,x,\text{and }y.$ However assuming the obvious I tried using this, but I am not able reduce it to ...
1
vote
2answers
655 views

bird traveling to a nest wants to save energy

This is a multiple choice question in one of tests I just wrote and I did not know the answer to it. I was just stuck on this during the test. It is a very weird question, one I find to be impossible. ...
3
votes
3answers
81 views

Prove that $(2-x)^nx^{n-1}$ decreases with $n$ for $0 <x<1$?

How can I show that: $$(2-x)^nx^{n-1}$$ is decreasing with $n$ when $0<x<1$? I think this is generally true, but specifically I am concerned with $n$ as an integer $\geq 2$ and showing that the ...