0
votes
0answers
8 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
1answer
48 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
113 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 ...
1
vote
2answers
161 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
25 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
49 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
46 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
244 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
74 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
106 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
24 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
43 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
41 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
135 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
83 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
266 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
121 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
167 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
50 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
388 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
74 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
120 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
106 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
42 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
54 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
167 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
84 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
118 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
79 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
135 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
338 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 $16$ ...
33
votes
4answers
862 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
189 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
579 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
80 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 ...
4
votes
8answers
2k views

Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for ...
3
votes
6answers
316 views

Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$

If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$. How to solve this one? (Here's the source of inspiration for the problem.)
1
vote
3answers
234 views

A “fast” way to ,find the maximum value of $(x^2) \times (y^3)$,if $3x+4y=12$ for $x,y \ge 0$

If $3x+4y=12$ $\forall x,y \ge 0$,the maximum value of $(x^2) \times (y^3)$ is $6 \times (6/5)^5$ $3 \times (6/5)^5$ $ (6/5)^5 $ $7 \times (6/5)^5$ How to approach this problem?I thought of ...
5
votes
5answers
861 views

Local minimum and maximum of the function

Can anyone help me to solve the following question? maximize and minimize the function $(10-x)(10-\sqrt{9^2-x^2})$ over $x\in[0,10]$ This is a high school question, so is there any simple trick help ...
3
votes
2answers
138 views

Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$

What is a good way to calculate max/min of $$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$ where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain ...
4
votes
1answer
408 views

Find the minimum for a trigonometric function

Find the local minimum of the following function: $$\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$$ I am wondering how can I simply this ...
1
vote
1answer
100 views

How can I find $\sup -\frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$ for $x_1 x_2 > 0$?

I need to find a constant $a$ such that for all $x_1 x_2 > 0$: $$a > - \frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$$ that is to say the supremum of the term on the right hand side. My question is how to ...
-2
votes
1answer
271 views

Polynomial problem

From http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext2_05.pdf: Suppose that $a$ and b are positive real numbers, and let $f(x)=\frac{a+b+x}{3(abx)^{\frac13}}$ for $x ...
1
vote
1answer
211 views

Finding the minimum value of a quadratic within a range

Given any quadratic equation of the form $y=ax^2+bx+c$, I want to find the minimum value for a specific range of $x$. My programmer brain can do it in a branchy, algorithmic way as follows, but is ...
2
votes
0answers
244 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...