0
votes
0answers
6 views

Constant such that Global Maximum is Particular Entry in Vector

Suppose I have $N\times 1$ vectors $X$ and $Y$, such that they are both are strictly increasing. I define a third vector as $Z\equiv f(X) + K g(Y)$ where $f$ and $g$ are arbitrary functions. I want ...
0
votes
1answer
79 views

Verify by Second Derivative Test

$$A(x)=2\sqrt{x^2-16}+\frac14\sqrt{68x^2-x^4-256}\;,\;\; (4 < x < 8)$$ of which the derivative is: $$a'(x)=\frac{2x}{\sqrt{x^2-16}}+\frac{136x-4x^3}{8\sqrt{68x^2-x^4-256}}$$ I first had to ...
2
votes
2answers
53 views

How to show a function has a global maximum without using derivatives

I want to show that $f(x) = |x|e^{-|x|}$ has a global maximum for some $x>0$. I don't want to use derivatives to do this. How can I do it?
0
votes
3answers
66 views

Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
4
votes
5answers
71 views

Local minimum global

Let $f:(a,b)\to\Bbb R$ be continuous. Assume that $f$ has a local minimum at some point $x_0$. Further assume that this is the only point where $f$ has a local extremum. Does it follow that $f$ has a ...
0
votes
1answer
52 views

How to solve for maximum area of a rectangle under a curve?

Having trouble with this optimization question and was hoping I could get some help with it. The function of the curve is $8^{-\frac{x}{5}}$. I would greatly appreciate a full explanation. I already ...
3
votes
0answers
55 views

Optimization problem (Sum of distances)

Given an ordered sequence $x_1 \leq x_2 \leq \cdots \leq x_n $ of length $n$ and a cost function $C(i) = \sum_{j}^{n}{\left|x_i-x_j\right|}$. The goal is to minimize the cost function. How do you ...
0
votes
3answers
64 views

Finding velocity in optimization problem

Given $s=-16t^2+192t+144$, what is the velocity when $s=0$? This is part of a larger optimization problem which I solved, except for this last part. The critical point occurs at $t=6$, so after ...
1
vote
2answers
288 views

Maximum of product of two functions

Let's have two (real continuous differentiable) functions such that $f(x)$ is bounded (from below and from above), positive ($f(x)>0$), and is strictly increasing ($f'(x)>0$, $\forall x$). ...
1
vote
1answer
25 views

Clarification about global extrema

Is everything in my statements $\textbf{(1)}$ and $\textbf{(2)}$ correct? A real valued function $f$ defined on a domain $X$ has a global maximum point at $x^{\bigstar}$ if $f(x^{\bigstar}) ...
1
vote
1answer
28 views

Maximization under constrains

I would like to maximize the function: $\frac{1}{2}\sum_{i=1}^{N}\lvert x_i-\frac{1}{N}\rvert$ under the constrains $\sum_{i=1}^{N}x_i=1$ and $0\le x_i \le 1$ $\forall i\in(1,...,N)$ I have done ...
1
vote
3answers
75 views

Determining domain interval for optimization problems

This example is from Paul's Online Notes for Calc I. You have $500$ feet of fencing material and you want to enclose a field with a fence. A building is on one side of the field (and so won't ...
0
votes
1answer
48 views

Bounded logarithmic function

I am trying to find any function that it grows logarithmically up to a certain point, and after that point it remains constant. Can anyone help me with that
1
vote
1answer
46 views

Help with local extrema of $f(x)=x^4-5x^2$

Find the coordinates of any local extreme points and inflection points of the function $f(x)=x^4-5x^2$ My try: Find critical points: $f^{\prime}(x)=4x^3-10x=0$ $f^{\prime}(x)=2x(2x^2-5)=0 ...
1
vote
0answers
25 views

Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
0
votes
2answers
77 views

Help finding local extrema of $f(x)=\frac{x}{\sqrt{2}}-3\sin\frac{x}{2}$

Find the local extrema of $f(x)=\dfrac{x}{\sqrt{2}}-3\sin\dfrac{x}{2}$ on the interval $0 \leq x \leq 2\pi$ $f^{\prime}=\dfrac{1}{\sqrt{2}}-3\cos \left(\dfrac{x}{2}\right) \left ( ...
0
votes
0answers
43 views

Find lower bound of function $x^{(1+x)/x}+(x+1)^{x/(x+1)}$

I would like to find a lower bound (linear) function of $$f(x)=x^{(1+x)/x}+(x+1)^{x/(x+1)}$$ where $x\in[3,+\infty[$? Any help would be most welcome. Thank you very much.
1
vote
2answers
126 views

Find lower bound of function $\frac{x}{x^{1/x}}$

Can someone help me finding a lower bound to the function $$f(x)=\frac{x}{x^{1/x}},$$ where $x\in[3,+\infty[$? I suppose that a lower bound function can be $y=x$ but I don't really know how to start, ...
0
votes
2answers
615 views

Difference between minimizing and maximizing functions

Could someone please explain the difference between minimizing and maximizing functions or give me some links to explain the difference in very very very simple terms? I have searched online and I ...
2
votes
2answers
183 views

Critical Values of a Function

I need to find the critical values of $h(t) = t^{3/4} - 2t^{1/4}.$ So I began by finding the derivative of the function and simplifying: \begin{align*} h'(t) &= (3/4)t^{-1/4} - (2/4)t^{-3/4} \\ ...
0
votes
0answers
26 views

Functional optimization problem with constraint

I have the following problem: $\min_{f(x): [a,b]\to \mathbb{R^+}} \max_{x \in [a,b]} x f(x)$ Subject to $\int_{a}^{b} f(x) dx = Z$, $\ f(x) \geq 0$, $\ 0 < a < b$, $\ f$ continuous. I ...
0
votes
2answers
58 views

Minimize $ab+bc+ca$ under three second degree constraints

As stated in the title, my problem is quite simple. Minimize $ab+bc+ca$ under these three constraints: $$ a^2+b^2=1 $$$$ b^2+c^2=2 $$$$ c^2+a^2=2 $$ I can brute force it, with some intelligence of ...
0
votes
1answer
19 views

Geometric interpretation of a critical point, i.e. of $q(t) := f(x + t(y-x))$.

So, I know what critical points are. But hear me out on the following notes I made: For $x,y\in \mathbb{R}^n$ we define $$q(t) := f(x + t(y-x)), $$ then $$q'(t)=\nabla f(x+t(y-x))^T(y-x).$$ Now, if ...
0
votes
1answer
25 views

Minimum of function

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with $f(x)=\dfrac{(x^4-2ax^3+3a^2x^2-2a^3x+a^4+9)}{(x^2-ax+a^2)}.$ Determine the minimum of the function, if we know, that $-2\leq a\leq2$, $a\neq0$. I ...
5
votes
1answer
191 views

Find lower bound of function

Can someone help me finding a lower bound to the function $$f(x)=\frac{x-1}{e^{-1}-xe^{-x^2}},$$ where $x\in[1,+\infty[$? Taking the derivative and then solve $f'(x)=0$ isn't analytically possible. ...
-3
votes
2answers
92 views

Guess the functional form of a graph

Can you guess the functional form of the following curve y is 0 at x= Infinite ; y is very small ( +ve near to zero) at x=0 Thanks and regards
3
votes
1answer
27 views

function with minumum in geometric mean

I have two real constants (in my case 3 and 15). I need a function that has minimum in the geometric mean and rises to infinity as I come closer to the end points. It only needs be defined on (3, 15). ...
1
vote
2answers
346 views

How to find minimum of sum of mod functions?

How to find minimum value of $$|x-1| + |x-2| + |x-31| + |x-24| + |x-5| + |x-6| + |x-17| + |x-8| + \\|x-9| + |x-10| + |x-11| + |x-12|$$ and also where it occurs ? I know the procedure for find answer ...
1
vote
3answers
864 views

Are absolute extrema only in continuous functions?

The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an ...
2
votes
2answers
133 views

Is there any way to find minimum without the use of derivatve?

The function is: $$\sqrt{(x+1)^2+\left(2x^2-\frac{1}{4}\right)^2}$$ It simplifies to: $$\sqrt{4x^4+2x+\frac{17}{16}}$$
1
vote
2answers
94 views

find maximum and minimum for any function

I'm writing an optimization algorithm thats supposed to find the maximum and minimum value of any given function. Whats the fastest numerical approuch to do so?
0
votes
0answers
183 views

De Jong's Fifth Function's Minimum?

What is the minimum solution to De Jong's fifth function, in the range $-65.536\leqslant x_1\leqslant 65.536, -65.536\leqslant x_2\leqslant 65.536$?
0
votes
0answers
60 views

Minimum of some functions

Denote $U=\{(x_1,x_2,...,x_n):0<x_j<1 (1\leq j\leq n),\sum_{j=1}^nx_j=1\}$. Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy: ...
0
votes
2answers
131 views

Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$

where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
0
votes
2answers
98 views

Minimizing The Cost

I have this exercise that I would like anyone to suggest the required steps in order to solve it A cylindrical can is to be made to hold $250 \pi\; cm^3$. Find the dimensions of the can that will ...
1
vote
1answer
187 views

Minimizing a linear combination of convex functions

Suppose a series of convex functions $f_1(x), f_2(x), f_3(x), ...$ is given (also, expressions for their derivatives $\nabla f_1, \nabla f_2,...$ are known). Now, suppose a function $g(x)$ is a linear ...
1
vote
1answer
53 views

General solvability at the stationary condition

Suppose a convex quadratic function $f(x)$ is given. To find a minimum of such function, one sets its derivative so zero, and solves for $x$. For instance, suppose that the result of differentiation ...
2
votes
4answers
260 views

How to calculate the maximum value of: $\frac{25x}{x^2+1600x+640000}$?

Wolfram says it's 800, but how to calculate it? $$ \frac{25x}{x^2+1600x+640000} $$
1
vote
2answers
350 views

Bell-shaped polynomial over a limited domain

The function $f(x) = e^{-x^2}$ has a bell-shaped peak at $x=0$ and then approaches an asymptote at $y=0$. I need to achieve a similar result, but with a polynomial function. I can use a series ...
2
votes
1answer
68 views

Root and sign of a complicated bivariate function

Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let $$ \Phi(p,i) := \frac{1}{2^p+1} + \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right), $$ where $\lg x$ is ...
1
vote
0answers
115 views

Properties of the sum or product of functions

I have $m$ functions of $n$ variables, $f_i(x_1,\dots,x_n)$ (where both $m$ and $n$ are finite), and I want to find the maximum (or the minimum) of: \begin{equation} G=\sum_{i=1}^{m}f_i(\bf{x}) ...
3
votes
1answer
167 views

Minimizing a function over two variables

Given two natural numbers $i$ and $p$ such that $0 < i \leqslant 2^p$, let $$ \psi(p,i) := p - \alpha + 1 - \frac{1}{2^p}\left((2^p+i)\lg(2^p+i) - i\lg i - i + \alpha - \frac{2^p}{i+1} - ...
0
votes
1answer
550 views

Maximizing and Minimizing a function

Let $f(x,y)$ be a function such that $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Now we have to maximize $f$ over $x$ and minimize it over $y$ $i.e.\ $ $$\underset{x}{\text{max}}\: ...
4
votes
5answers
106 views

Calculating maximum of function

I want to determine the value of a constant $a > 0$ which causes the highest possible value of $f(x) = ax(1-x-a)$. I have tried deriving the function to find a relation between $x$ and $a$ when ...
0
votes
2answers
290 views

General method to find inf, sup, maxs and mins of a function

Could someone explain how to find inf, sup, max and min values of a function (real-valued functions of real variable, generally continuous/differentiable, with some possible points of discontinuity)? ...
17
votes
2answers
647 views

Largest circle between $y=x^n$ and $y=\sqrt[n]{x}$

Something I have been wondering about for a while. Let us look at the area between $x^n$ and $\sqrt[n]{x}$ when $x\in [0,1]$. Where $n$ is a positive integer. Below is an image. With a given n, how ...
1
vote
0answers
164 views

Optimisation of Cost Functions with step functions

Hi I would like to know which algorithm is best suited to solve this Cost Minimisation problem: Total Cost = ...
2
votes
2answers
1k views

Looking for numerical methods for finding local maxima and minima of a function

In derivative, If $f'(x)$ is rising at $f'(x)$ = 0, there's a local minima in $f(x)$. If $f'(x)$ is falling at $f'(x)$ = 0, there's a local maxima in $f(x)$. If $f''(x)$ is ...
4
votes
2answers
880 views

Math notation for location of the maximum

My question is about notation. I have maximum of the function $f(x)$. This can be expressed as $\max(f)$ How can I express in compact form that $x_0$ is the location of that maximum.
6
votes
3answers
956 views

Lagrange Multipliers

I would like to find the extrema of the function $f(x,y)=x^2+4xy+4y^2$ subject to $x^2+2y^2=4$ using Lagrange Multipliers. Is it possible to get for the Lagrange multipliers the value zero? I don't ...