Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [maxima-minima]

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire ...

2
votes
2answers
110 views

Can critical points occur at endpoints? E.g. $f(x) = \frac{1}{x}$ at the interval $[1,4]$

Given that an extreme value can only occur at a critical point and in the following case $$f(x) = \frac{1}{x} \quad [1,4]$$ we definitely have two absolute extreme values (maximum and minimum), are $x ...
0
votes
2answers
53 views

Calculus Made Easy, Exercise 9.3, maximizing area of rectangle

STATEMENT: A line of length $p$ is to be cut up into $4$ parts and put together as a rectangle. Show that the area of the rectangle will be a maximum if each of its sides is equal to $1/4p$. PROBLEM: ...
1
vote
3answers
38 views

Ask about global maximum and global minimum?

The temperature distribution in a metal rod given by the following function of the position $x \in \mathbb{R}$: $$T(x) = \frac{1 + 2x}{2 + x^2}$$ What is the maximal and minimal temperature in the ...
0
votes
1answer
45 views

Finding the perimeter of the required rectangle

A rectangle is inscribed within another rectangle of dimensions 6 units by 8 units. The rectangle has been inscribed in it with its vertices on the sides of the rectangle in such a way that if the ...
0
votes
2answers
34 views

Find the index of the largest element in a sequence $x_n = (\alpha + \beta)^{-n} \sum_{k=0}^nc^{n-k}\beta^kx_{n-k}x_k$.

As in the title i want to find the largest element of the given sequence. Excuse me for the vague title, MSE limits the title to 150 symbols. To give you some context here is the full problem ...
0
votes
0answers
59 views

Is there any way to solve this optimization problem better than exhaustive search?

Here is the optimization problem: For the function $$ f(x_1,x_2;a_0,b_0)=\\\small\cases{\frac{1}{2}\left[x_1+x_2-x_1x_2+\left(\frac{-1+b_0(1-a_0)}{a_0}x_1+1\right)\left(\frac{-1+b_0(1-a_0)}{a_0}x_2+...
0
votes
2answers
73 views

Absolute minimum of $f(x,y,z)=xy+yz+zx$ on $x^2+y^2+z^2=12$

I'm trying to find the absolute minimum of $f(x,y,z)=xy+yz+zx$ on $g(x,y,z)=0$ where $g(x,y,z)=x^2+y^2+z^2-12$. I'm using the method of Lagrange multipliers: solving the system $$\nabla f=t\nabla g\\...
0
votes
0answers
28 views

Global maximum for a two times continuously differentiable function

Let $M:=\{(x,y)\in \mathbb R^2:x^2+y^2<1\}\setminus\{(x,0)\in \mathbb R^2, x\in \mathbb R\}$ and $f:\mathbb R^2\rightarrow \mathbb R$ be two times continuously differentiable. I want to know ...
0
votes
1answer
40 views

Find the maximum constant such that the inequality

Let $a;b>0$. Find the maximum constant such that the inequality $$\frac{1}{a^2+b^2}+\frac{1}{a^2}+\frac{1}{b^2}\ge \frac{8+2k}{\left(a+b\right)^2}$$ Let $a=1$ then we have: $-\frac{k-1}{2a^2}\ge 0\...
4
votes
1answer
238 views

What can you do if the higher-order derivative test is inconclusive?

The second-derivative test states that if $x$ is a real number such that $f'(x)=0$, then: If $f''(x)>0$, then $f$ has a local minimum at $x$. If $f''(x)<0$, then $f$ has a local maximum at $x$...
1
vote
1answer
112 views

Is $f(x) = x \sqrt{4-x}$ decreasing at $x = 4$?

I am solving a single variable calculus problem and it asks me to determine the decreasing and decreasing intervals of the function $f(x) = x \sqrt{4-x}$. It's pretty clear that from $\left] -\infty , ...
-1
votes
2answers
45 views

Find the greatest value of the function $f(x) = x^4 - 6bx^2 + b^2$ [closed]

Find the greatest value of the function $f(x) = x^4 - 6bx^2 + b^2$ on the interval $[-2, 1]$ depending on the parameter $b$.
3
votes
3answers
63 views

Find the least value for $\sin x - \cos^2 x -1$

Find all the values of $x$ for which the function $y = \sin x - \cos^2 x -1$ assumes the least value. What is that value? At first I found the first derivative to be $y' = \cos x + 2 \sin x \cos x$. ...
0
votes
0answers
57 views

Maximum preserved under monotonic mapping

For some strictly non-negative function $f\,\colon \mathbb{R} \to \mathbb{R}^+$, and a monotonically increasing function $g\,\colon \mathbb{R} \to \mathbb{R}^+$, is there a name (theorem or lemma name?...
1
vote
4answers
223 views

Minimizing RSS by taking partial derivative

I am learning about linear regression, and the goal is to find parameters $\beta$, that minimize the RSS. My textbook accomplishes this by finding $\partial \text{ RSS} /\partial \beta = 0$ However, I ...
2
votes
5answers
121 views

Minimum value of $\frac{x^4+5x^2+7}{x^2+3}$

Minimum value of $$f(x)=\frac{x^4+5x^2+7}{x^2+3}$$ we have $f(x)$ as $$f(x)=(x^2+3)+\frac{1}{x^2+3}-1$$ Now by $AM \gt GM$ we have $$(x^2+3)+\frac{1}{x^2+3} \gt 2$$ But equality cannot occur ...
4
votes
6answers
193 views

Maximum minus minimum of $c$ where $a+b+c=2$ and $a^2+b^2+c^2=12$

Let $a,b,$ and $c$ be real numbers such that $a+b+c=2 \text{ and } a^2+b^2+c^2=12.$ What is the difference between the maximum and minimum possible values of $c$? $\text{(A) }2\qquad \text{ (B) }\...
2
votes
1answer
37 views

Exchanging max, log and absolute value

Let $N \in \mathbb{N}$, $f,g: [N] \to (0,1]$. It is a consequence of the triangle inequality that : $$\left|\max_{n \in [N]} f(n) - \max_{n \in [N]} g(n)\right| \leq \max_{n \in [N]} \left| f(n) - g(...
0
votes
1answer
44 views

Maximum of a linear function in a set that is convex but not compact

Let $f: V\to \mathbb{R}$ be a linear function defined on some real vector space $V$. It is known that, if $U$ is a convex and compact subset of $V$, then $f$ attains its maximum at some extreme point ...
0
votes
0answers
16 views

Local Extreme Values over subdomains

I consider a $f(x,y)$ function which is continuous on the domain $X=[0,1]\times [0,1]$. I am trying to find local extreme values of the $f(x,y)$ function by splitting the whole domain $X$ into small ...
0
votes
2answers
33 views

Maximum of a Product

At what value of $x$ does the maximum of $\ln(\prod_{i=1}^{k} xe^{-xa_{i}})$ occur? My initial calculation gave me $\sum_{i=1}^n a_i$. Can someone double check me?
4
votes
1answer
122 views

How to get a recurrence relation from an expression with maximum

There is a certain function $F(r,s)$ (where $r\geq -1$ and $s\geq 0$ are integers) that satisfies the following relation: $$ \max\big[2 F(r,s) - F(r-1,s) - F(r-1,s-1)~~~,~~~F(r+1,s+1) - F(r-1,s)\big] ...
2
votes
0answers
42 views

Solutions of two linear programming

Let $\beta\equiv (\beta_0, \beta_1)\in \mathcal{B}\subset \mathbb{R}^2$ with $\mathcal{B}$ compact. $\beta$ is a known vector of parameters. Let $P_0, P_1, F_0, F_1$ be know parameters, each in $[0,1]...
0
votes
1answer
49 views

Find critical points (minima) from gradient (2D vector)

I'm trying to find critical points (minima) for a gradient in 2D space. The following are partial derivative and its polynomial equation: To find critical points I will need to set gradient to 0 and ...
1
vote
3answers
66 views

How to find a minimax solution to a set of linear inequalities?

Let's consider the following linear inequalities: $$a - 10 \leq b \leq a - 7 \\ b + 3 \leq c \leq b + 6 \\ c + 3 \leq d \leq c + 6 \\ d + 3 \leq e \leq d + 6$$ Is there a way to find a ...
0
votes
1answer
9 views

Minimum of random and constant scaled with random variable

Suppose that $X$ and $Y$are independent random variables, while $c \geq 0$ is a constant. More specifically, $X$ is the $n$-fold convolution of $Y$. Is it then correct to say the following? $\min(Y,c)...
0
votes
1answer
25 views

Integration of a function that is the extreme value of a function having three variables

Consider $$\phi(a,b,t) = a^4 - 5a^2 +b^2 + 5t^2 -4bt -2t + \frac{33}{4}$$ where $a,b,t ∈ R$.Given that $f(t)$ and $g(b)$ are the minimum values of $\phi(a,b,t)$. Based on this statement there are two ...
1
vote
0answers
30 views

properties of $\min(x_1…x_n)$

I want to take measurements of algorithm performance. I have two algorithms A and B that run one after the other (composition) I want to measure how well the composition of algorithms is better than ...
1
vote
2answers
61 views

Any way to mathematically express the set of all argmin(f(x))?

my question is: I would like to express in a mathematical way the first $argmin(f(x))$. The function $argmin$ returns the argument for the global minima, but, there is any way to express the set of ...
0
votes
0answers
24 views

An absolute minima can be considered local?

local minimum value of f if $f(c) \leqslant f(x)$ when x is near c. I have reading comprehension problem... By the definition above can an absolute minimun also be included as local?
0
votes
5answers
110 views

Deriative of ${\frac{(\ln x)^{2}}{\sqrt{x}}}$ is not correct [closed]

Using the chain and quotient rules, I get: $${\frac{2*\ln x}{x*\sqrt{x}}-\frac{(\ln x)^{2}}{2*\sqrt{x}}}$$ Answer:$(0,0)$ - min, $(1,0)$- min, $(e^4$,$\frac{16}{e^2})$ - max Can anyone please ...
1
vote
1answer
46 views

Maximizing the value of a two variable function along any curve

I read that, of all the points on an origin-centered circle in the x-y plane, the function $z=ax+by$ is maximum (or minimum) at the point where $\frac{x}{y}=\frac{a}{b}$ I think this is too specific. ...
3
votes
2answers
102 views

Ways to find $\min[f(x)]$ where $f(x) = (x-1)(x-2)(x-3)(x-4)$ without using derivatives

As title states I need to: Find $\min[f(x)]$ where $f(x) = (x-1)(x-2)(x-3)(x-4)$ without using derivatives Since I i'm restricted to not use the derivatives I've started to play with the function ...
0
votes
1answer
29 views

How do i know $x(t)$ must be the largest or smallest value when I let $x'(0)=0$,and substitute the $s_k$ value back into this formula?

How do i know when i let $x'(0)=0$,x(t) must have the largest or smallest value,For example,let $x(t)=at^2+bt+c$,when we differential it with t and set the differential formula be equal to zero,and we ...
1
vote
1answer
40 views

Find global minima of vector valued quadratic equation [closed]

I have the following equation, with 5 $\mathbb{R}^3$ vectors A, B, C, D, and P, and a scalar, $t$; $-At^3 + 3Bt^3 + Ct^3 - 3Dt^3 + 3At^2 - 6Bt^2 + 3Dt^2 - 3At + 3Bt + A - P$ I'm trying to find the ...
0
votes
2answers
21 views

The set of values of $a$ for which the function does not posses critical points

The set of all values of '$a$' for which the function, $f(x)=(a^2-3a+2)(\cos^2{x/4} - \sin^2{x/4}) + (a-1)x + sin1$ does not posses critical points is: I first differented it to find $f'(x)$, then ...
0
votes
0answers
21 views

minimization problem of integral functional for a given function

Let $\Omega \subset \mathbb{R}^{n}$ ($n\geq 2)$ be a bounded domain satisfying an exterior ball condition for any $\xi_{0} \in \partial\Omega$ and $\forall x \in \overline{\Omega},\, u_{\xi_{0}}(x) = \...
1
vote
0answers
26 views

max(y) and max(x) for $1 \geq xy\lceil{\log_2{x}}\rceil\times\frac{1}{100}$

$1 \geq x\times\lceil{\log_2{x}}\rceil\times y\times\frac{1}{100}$ $x \in \mathbb{N}$, $x \geq 2$ $y \in \mathbb{Q}$, $y \geq 0$ now I can declare $max_y(x) = \frac{1}{x\times\lceil{\log_2{x}}\...
0
votes
0answers
40 views

Model the following scenario as a multivariable equation with a set of constraints and find the maxima

Greg wants to maximize his credit card rewards and has narrowed his options down to 3 credit cards. However, Greg only wants to apply for 2 new credit cards to avoid hurting his credit score. Based on ...
2
votes
1answer
46 views

Question on Local Maxima and Local Minima

Find the set of all the possible values of $a$ for which the function $f(x) = 5 + (a-2)x + (a-1)x^2 - x^3$ has a local minimum value at some $x < 1$ and local maximum value at some $x > 1$ The ...
-1
votes
1answer
81 views

Find range of the function $f(x)=\frac{\left\{x\right\}}{1+(\lfloor x\rfloor)^2}$

Find range of the function $f : \mathbb{R} \to \mathbb{R}$ $$f(x)=\frac{\left\{x\right\}}{1+(\lfloor x\rfloor)^2}$$ My try: Obviously range contains zero, since for integers $\left\{x\right\}=0$ ...
0
votes
1answer
39 views

Local minimum of an analytic function

This is a follow-up to a previous question of mine. I know that any local minimum $x_0$ of a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. ...
1
vote
3answers
60 views

Minima of $f(x)$

The miniumum value of $\left(1+ \dfrac{1}{\sin^n \alpha}\right)\left(1+ \dfrac{1}{\cos^n\alpha}\right)$ is? Attempt: I expanded the brackets and then differentiated and set the derivative equal to ...
0
votes
0answers
49 views

Inflection point could not be a local extreme point? [duplicate]

Why is the statement "An inflection point can not be a local extreme point?" wrong? Isn't a local extreme point either a max or a min only? What is wrong here?
1
vote
0answers
28 views

Partial minimization of function over variables [duplicate]

In Boyd and Vandenberghe's textbook on Convex Optimization, is claimed that: We always have $$ \underset{x,y}{\text{inf}\; f(x, y)} = \underset{x}{\text{inf}}\;\tilde{f}(x) $$ where $$ \tilde{f}(...
3
votes
2answers
38 views

Local minimum has neighbourhood with positive semi-definite Hessian?

I know that any local minimum $x_0$ of a (twice continuously differentiable) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. Can we say ...
0
votes
1answer
36 views

What theorem is used to justify the existence of a minimum of $f$ on $M$

Let $M:=\{(x,y,z) \in \mathbb R^{3} : x^2+y^2+z^2\geq 36\}$ and $f: \mathbb R^{3} \to \mathbb R$, $f(x,y,z)=\frac{x^2}{2}+\frac{y^2}{4}+\frac{z^2}{6}$ Justify why $f|_{M}$ takes on a minimum The ...
0
votes
3answers
57 views

Minimum value - Extension of Triangle Inequality? [duplicate]

What is the minimum value of $|x-1| + |x-2| + |x-3| .... + |x - k + 1| + |x-k|$ equal to? I suppose it depends on whether or not $k$ is even or odd. I was able to solve for $k = 3$ (three terms) ...
0
votes
0answers
65 views

Minimum of a function.

Determine the minimum value of the expression $$x^2+y^2+5z^2-xy-3yz-xz+3x-4y+7z$$ where x , y and z are real numbers. My Solution : Let $f(x,y,z)=x^2+y^2+5z^2-xy-3yz-xz+3x-4y+7z$ I calculated the ...
0
votes
1answer
20 views

Evaluating an extremal value if the hessian matrix has at least one eigenvalue which is zero

$$f(x,y) = 2x^4-3x^2y + y^2$$ We want to find all extremal values: $$df(x,y)=(8x^3-xy,-3x^2+2y)\overset{!}{=}0 \quad \Rightarrow \quad p=(0,0)$$ $$H_f(x,y)=\begin{pmatrix}24x^2-6y& -6x\\-6x &...