Mathematics Stack Exchange
Mathematics 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 domain of a function (the global or absolute extrema).

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 domain of a function (the global or absolute extrema).

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

A real-valued function $f$ defined on a domain $X$ has a global (or absolute) maximum point at $x^∗$ if $f(x^∗) \ge f(x)$ for all $x$ in $X$. Similarly, the function has a global (or absolute) minimum point at $x^∗$ if $f(x^∗) \le f(x)$ for all $x$ in $X$. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

3734 questions
0
votes
2 answers

Proof of Schwarz lemma 5

I dont understand the following proof of Schwarz lemma: Schwarz Lemma In the last section it says, "Moreover, suppose that $|f(z)| = |z|$ for some non-zero $z$ in $D$, or $|f'(0)| = 1$. Then, $|g(z)| = 1$ at some point of $D$". Why does $g(z)$ have…
Steven33
  • 117
0
votes
1 answer

Question on Maximum and minimum possibilities

In a zoo with 100rabbits ,with three kinds of rabbit 1kg,2kg,5kg. There are minimum of 10 rabbits of each kind and maximum of 60rabbits of each kind. If 40 rabbits weighing a total of 148 kg are transferred from one zoo to another,then the remaining…
Rahul
  • 21
0
votes
1 answer

Is this equation solvable (by hand) and what do you call this equation?

Is this equation solvable (by hand) and what do you call this equation? $a = x - \min(b\, x, c)$ $a, b, c$ are known. We need to solve for $x$. If it is, and I can get guidance as to what this is called, and where there are resources to find out how…
RyanH
  • 103
0
votes
2 answers

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 I tried to find out the condition where it…
Iceberry
  • 634
0
votes
0 answers

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 partial derivative equations as…
Math Tise
  • 657
  • 1
  • 5
  • 11
0
votes
2 answers

Minimum of a function $f(x,y)=x^2+y^2+\alpha/(xy)$

Let $\alpha \in \mathbb{R}^{*+}$ and $\left(x,y\right) \in \left(\mathbb{R}^{*+}\right)^2$, I have the function $f$ given by $$ f\left(x,y\right)=x^2+y^2+\frac{\alpha}{xy} $$ I've found that there exists one critical point which is $\displaystyle…
Atmos
  • 7,369
0
votes
3 answers

What is the maximum product of two numbers whose addition is 17 where the two numbers are integers?

It is very obvious that the numbers are 8.5 and 8.5 if it is not integer x+y=17 y=17-x Let t=xy=x(17-x) Max of t is 72.25 If it is integer the answer is 9 and 8 and the maximum product is 72 My question is to know is there any mathematical method to…
Yaro
  • 57
0
votes
1 answer

Reducing the function from two variables to one

$$S=\max\left\{{|x-y|\over1+x+y}:0
Anvit
  • 3,379
0
votes
1 answer

What will be max$\{0,a+b\}$?

I know max$\{0,a+b\}=\mbox{max}\{0,a\}+\mbox{max}\{0,b\}$, whenever $a$ and $b$ share same sign.What happens if they have different sign? Can we have some identity?
Abhinav Jha
  • 862
0
votes
1 answer

Extremum of $\frac{ax+by+c}{\sqrt{(1+x^2+y^2)}}$

Find the extremum of $$\frac{ax+by+c}{\sqrt{1+x^2+y^2}}$$ where $a^2+b^2+c^2>0$. I managed to find critical point - ($\frac{a}{c}$,$\frac{b}{c}$) if $c \neq0$, and no points otherwise. Value of function at this point looks great:…
Some
  • 19
0
votes
2 answers

Then determine the relative and absolute extrema

Consider the function $ f(x)=\begin{cases} 2x^3-3x^2-12x+9 , & x \leq 3 \\ 9x-x^2-18 , & x>3 \end{cases} $ over the interval $ \ [-2,5]\ $ Then determine the relative and absolute extrema . Also find the interval where $ \ f(x) \ $ is increasing or…
MAS
  • 10,638
0
votes
1 answer

How to find out the minimization of function of infinite variables?

Let we are given with a function of infinite variables $$f(x_1, x_2, \ldots) $$ which is differentiable. Now if we know that minimum of the above function exist, then can we apply the same condition i.e. $$\frac{\partial f}{\partial x_i} = 0…
PAMG
  • 4,440
0
votes
1 answer

What are the critical points of $f(x,y,z)=x^2+ay^2+z^2-4xy, \ \ where \ \ a \in \mathbb{R} $ .

What are the critical points of $ f(x,y,z)=x^2+ay^2+z^2-4xy, \ \ where \ \ a \in \mathbb{R} $ . Answer: The critical points are obtained by $ f_x=0, \\ f_y=0 , \\ f_z=0 \\ $ These gives $ x-2y=0 , \\ ay-2x=0, \\ z=0 \\ $ This gives $ \ (0,0,0)…
MAS
  • 10,638
0
votes
1 answer

For maximum value of $ P(t) \ $

Find the maximum value of $ \ P(t) \ $ given by $P(t)=0.0000000 219 t^4-0.0000 167 t^3+0.00155t^2+0.002t+0.23 \ $ For what value of $ \ t\ $ , $ \ P(t) \ $ is maximum ? Answer: For maximum value of $ P(t) \ $, $ P(t)'=0 \\ \Rightarrow 4 \times…
MAS
  • 10,638
0
votes
0 answers

Determine all values function reaches local max

Determine all $x > 0$ where function $f(x) = 1 + \int_{-x}^{x}\cos(t^2) dt $ reaches local maximum. I took the derivative of this function and got $f^\prime(x) = 2\cos(x^2)$ but further i put this equal to zero but can't find out the values for…
Afraz Salim
  • 1
1 2 3
…
9 10