-1
votes
0answers
18 views

How to find max and min bounds of a uncertain function

First I would like to say that I have searched the for uncertain fitting, robust fitting, linear optimization, convex optimization, etc. But I'm lacking the knowledge to solve this problem, and I need ...
1
vote
1answer
40 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
0
votes
1answer
25 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
0
votes
1answer
23 views

Objective function with two variables

A factory produces jointly two articles, and it has the problem to decide their prices in order to maximize the monthly income, knowing that the demand d1 (in hundreds of units) of the first article ...
2
votes
2answers
50 views

If a continuous function on $\mathbb R$ satisfies $f(x)\ge x^2$, it attains its minimum

This question is from my homework and I don't know how to prove it. Let $f(x)$ be a continuous function at $\mathbb{R}$. prove that if $f(x)\geq x^2$ to every $x$ in $\mathbb{R}$, then $f(x)$ ...
3
votes
0answers
53 views

Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
0
votes
1answer
34 views

Global/local optima for this function

I have the following function $f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$ where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant I want to find global/local maxima for this. ...
-1
votes
3answers
34 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
24 views

How to find the smallest value by using Lagrange multiplicators?

Let $a$, $b$ and $c$ be positive constants. How one can find the smallest value of the sum of three numbers $x_1$, $x_2$ and $x_3$ at the surface $\dfrac{a}{x_1}+\frac{b}{x_2}+\frac{c}{x_3}=1$ by ...
0
votes
0answers
9 views

Proper name for the problem (finding optimal discrete function)

Given a set $D = \{d_1, d_2, ..., d_N\}$, a set of some subsets of $D$, $D^\ast$ and a set of classes, $C = \{c_1, c_2, ..., c_M\}$, I want to find function, that maps a sequence $({d_i}_1^\ast, ...
0
votes
1answer
38 views

A function with positive Hessian at a critical point, without having a minimum there

I have a problem with a little instance: $f(x,y) = \begin{cases} (x^4-3x^2y^2+y^2)/(x^2+y^2) & otherwise \\ 0 & \text{(x,y)=(0,0)} \end{cases}$ This is a example of a function which ...
0
votes
3answers
42 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
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 ...
2
votes
1answer
38 views

Second derivative test inconclusive, all derivatives are 0, moving critical point to origin, no result?

Here is a function $f(x,y)=x^4 + 6x^2y^2 + y^4 -4x^3 - 12xy^2 + 6x^2 + 6y^2 - 4x + 1$. I've happily proved that $(1,0)$ is a critical point for that function. Now I'd like to decide whether is it a ...
1
vote
2answers
118 views

The minimum of $\int_a^b |f(t)-x|\,\mathrm{d}t$ over $x\in \mathbb {R}$

I found this statement in a book, given without proof and left as an exercise to the reader. Theorem: Suppose $f$ is a continuous and strictly increasing function on $[a,b].$ Let $m=(a+b)/2$. ...
0
votes
1answer
99 views

How do you minimize “hinge-loss”?

A lot of material on the web regarding Loss functions talk about "minimizing the Hinge Loss". However, nobody actually explains it, or at least gives some example. The best material I found is here ...
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
0answers
6 views

Steps in Anlaysing Surface Plot

I have two vectors ${\bf x}$, ${\bf y}$ such that $0 \leq x_i, y_j \leq 1$ and a third vector ${\bf z}$ where $z_i \in \{0,1\}$. Each row of the vectors represents an 'observation' and I am trying to ...
0
votes
3answers
32 views

Why is max($\frac{2}{||w||}$)= min($\frac{1}{2}$)($||w||^2$)?

I was watching a video on machine learning. The instructor says that maximizing ($\frac{2}{||w||}$)is difficult (why?) so instead we prefer to minimize $\frac{1}{2}||w||^2$. $w$ is a vector. How are ...
0
votes
1answer
18 views

Proving a Function is Less than a Value

I'm working on a multipart problem and I've been asked to prove that a value W < 0.5. I worked out W and reached $$W = \frac{\sqrt{X1*X2*Y1*Y2}}{X1*Y1+X2*Y2+X1*Y2} $$ but I'm not sure how to ...
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
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
98 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
82 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
136 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, $h(x) = \frac{f(x)}{g(x)}$, ...
3
votes
5answers
82 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
72 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
58 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
103 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
330 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
27 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
29 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
190 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
77 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
75 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
31 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
87 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
48 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
145 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
1k 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
283 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
27 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
310 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
109 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
522 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
1k 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 ...