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).
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when $\underset{x\in I}{\max}|(x-x_0)(x-x_1)|$ is minimum
I'm trying to find out for what values of $x_0$ and $x_1$, $\underset{x\in I}{\max}|(x-x_0)(x-x_1)|$ becomes minimum for $I=[-1,1]$. Note that we only look at the function diagram in $[-1, 1]$. I ...
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Geometric interpretation of maximizing utility functions
So this was the problem I was trying to solve, and I got stuck in part (b).
When you total differentiate $U(x,y)=0$, we can see that $Uxdx+Uydy=0$, which means $dy/dx=-Ux/Uy$
However, I am struggling ...
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1
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Circles1 with maximum and minimum values [closed]
I have a question regarding the maximum and minimum values of $y-3x+4$ for which point $(x,y)$ moves along circle $(x-2)^2+y^2=1$.
I can calculate the question but I do not know where to locate the ...
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Calculating maximum values where the function is not defined
OK, this looks like nonsense, and, in a way, it is.
Looking for a maximum of $f(x)=\frac1{x^2}$ we can't have one, because the obvious, infinite maximum would be at $x=0$ where the function is not ...
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55
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Finding min value of exponential expression.
Let $x,y, z$ be reals such that $x^2+y^2+z^2=3$. What is the minimum value of $2^{1/x}+2^{1/y}+2^{1/z}$?
First I thought taking $x,y,z$ as $-1$ which leads to $3/2$. However by trial and error, if I ...
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For $x,y,z$ positive real numbers $3x+4y+7z=1$. Find minumum integer value of $1/x+1/y+1/z$
For $x,y,z$ positive real numbers $3x+4y+7z=1$. Find minimum integer value of $1/x+1/y+1/z$
My solution like this:
Using Cauchy-Schwarz inequality $$(3x+4y+7z)(1/x+1/y+1/z)\ge (\sqrt{3}+2+\sqrt{7})^2$$...
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If $\Delta_3 u>0$ on $D$, prove $\max_{x\in D}u(x)=u(x_0)$ for some $x_0\in \partial D$
This is a problem in a PDEs course I am helping a student with, and I would like to take this opportunity to verify my solution. I will also entertain alternative/easier approaches, because I suspect ...
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What's the minimum and maximum number of right exterior angles in a convex octagon, acute exterior angles, obtuse exterior angles? [closed]
What's the minimum and maximum number of right exterior angles in a convex octagon, acute exterior angles, obtuse exterior angles?
A convex polygon is one in which none of the interior angles are ...
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If $f(x)= \sum\limits_{k=1}^{m}(x-k)^4$ such that $m>1$ Then prove that f(x) has a unique minima and find it.
If $$f(x)= \sum\limits_{k=1}^{m}(x-k)^4$$
such that $m>1$
Then prove that $f(x)$ has a unique minima and find it.
My Attempt:-
$f'(x) = 4[ (x-1)^3+(x-2)^3+....+(x-m)^3] $
At a point of minimum $f'(...
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3
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Maximum value of $(1-a)(1-b)+(1-p)(1-q)$
Given that real numbers $a,b,p,q$ satisfy $$a^2+b^2=p^2+q^2=2$$
Find the maximum value of $E=(1-a)(1-b)+(1-p)(1-q)$.
My try:
I have chosen $a=\sqrt{2}\sin x, b=\sqrt{2}\cos x, p=\sqrt{2}\sin y, q=\...
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Finding $\small{\min\limits_{ab+bc+ca=1}\sqrt{a+2}+\sqrt{b+2}+\sqrt{c+2}- \sqrt{2-abc}.}$
Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Find the minimal value of expression
$$P=\sqrt{a+2}+\sqrt{b+2}+\sqrt{c+2}- \sqrt{2-abc}.$$
By $a=b=1;c=0$ I get $P=2\sqrt{3}$ so we ...
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Prove that we have a local maximum at $0$ for this function involving a little-o estimate and a symmetric positive-definite matrix
Let $f:\mathbb R\times\mathbb R^n\to\mathbb R$ be the function defined as
$$f(y, x)= G(y, x) -\frac{1}{4} M(y)x\cdot x,$$
where $G\in C^2(\mathbb R\times\mathbb R^n, \mathbb R)$ and $M\in C(\mathbb R, ...
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1
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Does $f(t)+sf\left(\frac{r-t}{s}\right)$ have local extrema at $r/(s+1)$ and $-r/(s-1)$ for $s > 1$, $r \neq 0$.
Let $f:\mathbb{R}\mapsto [-1,1]$ be a nondecreasing function, such that
$f(-t) = -f(t)$, i.e. it's odd.
It is convex on $(-\infty, 0]$ and concave on $[0, \infty)$
Is it then true that for $s > 1,...
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Does the converse of the first derivative test hold for isolated extrema?
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $c \in \mathbb{R}$ be a critical point of f. Let f be differentiable on an open interval containing c, except possibly at c.
Can we say ...
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Find the maximum difference between the limits of integration
Let $ C=\int_{a}^{b} (7x-x^2-10) dx $ where $a<b$
Determine the maximum value $(b-a)$ can assume if $C=0$
This question has troubled me for some time, and I would like some help to solve this ...
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Strictly positive absolute maximum of a continuous function vanishing at infinity
Let $f:\mathbb R\to\mathbb R$ be a continuous and not identically zero function such that $$\lim_{x\to\pm\infty} f(x)=0.$$
Under which other extra assumptions I can say that $f$ has an absolute ...
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Minimal area of triangle in coordinate system
Given a line in the first quadrant of the cartesian plane tangent to the unit circle, consider the triangle formed by this line and the positive $x$ and $y$ axis. Prove that the minimal area of this ...
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3
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Finding minimum in functions of 2 variables
Question : Find all (x,y), x,y> 0 for which f(x,y) is minimum where
$f(x,y) = \frac{x^4}{y^4}+\frac{y^4}{x^4}-\frac{x^2}{y^2}-\frac{y^2}{x^2}+\frac{x}{y}+\frac{y}{x}$
My attempt:The values of (x,y) ...
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3
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Find the minimum value of $f(a,b) = 5 - 2ab -4(a - b)$ subject to $a^2+b^2 = 1$ without calculus
While tutoring my pre-cal tutees, I spontaneously created a problem about minimum value that involves a unit circle and a straight line $y = x - 2$. Specifically, I was asking my tutees the following ...
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How to write $\min\{a,k\}+\min\{b,k\}$ as one min?
I want to simplify a solution to a problem I have where I used to write
$$\min\{a,k\} + \min\{b,k\} \leq \min\{c,k\} + \min\{d,k\}.$$
I am wondering: is there a more simplified way of writing this ...
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How do you determine local minima from 2D plot?
I'm trying to find how many local minimas the function have in the defined region but I don't quite understand how you can tell which one is a local minima from a 2D plot. From videos I have seen, it'...
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Find all possible values of $H(x,y,z,t) = \frac{x}{t+x+y}+\frac{y}{x+y+z}+\frac{z}{y+z+t}+\frac{t}{z+t+x}$ if $x, y, z, t > 0$.
If $x, y, z, t > 0$, find all possible values of $H(x,y,z,t) = \frac{x}{t+x+y}+\frac{y}{x+y+z}+\frac{z}{y+z+t}+\frac{t}{z+t+x}$.
How I think this can be solved:
First off, note that $H$ is an ...
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Optimization of a linear PDE and operator formalism
Consider the reaction-diffusion PDE:
$$\partial_t m(x,t) = f(x,t) \, m(x,t) + K \, \partial_x^2 m(x,t)\,, \qquad
m(x,t=0) = \frac{\exp\lbrace-x^2/(2\sigma_0^2)\rbrace}{\sqrt{2\pi\sigma_0^2}}$$
for $x\...
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Finding The Least Integer Value
$\sqrt{x^{2}+9}+\sqrt{\left(y-x\right)^{2}+4}+\sqrt{\left(9-y\right)^{2}+49}$
(x and y is not zero and different from each other.)
How can I find the least integer value of that kinda expression above ...
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What is a notation for the histogram's bar corresponding to the "leftmost local maxima"?
Question. What is a notation for the histogram's bar corresponding to the "leftmost local maxima"?
Maybe what proposed in Peak / local maxima notation can be adapted to denote the bar ...
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Prove $f(a,b,c)\le f\left(\frac{a+b}{2},\frac{a+b}{2},c\right)$
Given non-negative real numbers $a,b,c$ which sum is $2.$ Assume that $c=min\{a,b,c\}.$
Denote $$f(a,b,c)=\frac{\sqrt{2ab+7}+\sqrt{2bc+7}+\sqrt{2ca+7}}{\sqrt{abc+9}}.$$
Prove that $f(a,b,c)\le f(t,t,c)...
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Minimize $P=\dfrac{\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}}{\sqrt{ab+bc+ca+6}},$ if $a+b+c+abc=4.$
Problem. Let $a,b,c\ge 0: a+b+c+abc=4.$ Find minimal value of $P$ $$P=\frac{\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}}{\sqrt{ab+bc+ca+6}}.$$
Source: Vo Quoc Ba Can.
My attempt:
Set $$P(a,b,c)=\frac{\sqrt{a+1}+...
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Finding $\small{\max\limits_{ab+bc+ca=1}\sqrt{\frac{1}{a+1}+\frac{1}{b+1}}+\sqrt{\frac{1}{a+1}+\frac{1}{c+1}}+\sqrt{\frac{1}{c+1}+\frac{1}{b+1}}.}$
Let $a,b,c\ge 0: ab+bc+ca=1.$ Find the maximum $$P=\sqrt{\frac{1}{a+1}+\frac{1}{b+1}}+\sqrt{\frac{1}{a+1}+\frac{1}{c+1}}+\sqrt{\frac{1}{c+1}+\frac{1}{b+1}}.$$
By denote some specific value, I think ...
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Is there any kind of distributive law for $c\cdot\max(a,b)$, allowing both signs of $c$?
Notation: For any two numbers $a$ and $b$, let the maximum be $a\sqcap b$, and let the minimum be $a\sqcup b$. (No, my symbols aren't upside-down. Compare this with floor notation; $\lfloor a\rfloor\...
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Minimizing Absolute Value
Let a, b, c be three nonzero integers satisfying 7a + 11b + 13c = 0. What is the least possible value of |a| + |b| + |c|?
I tried graphing this on the x-y-z plane but that didn't help much as a, b, ...
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Finding Equation of Quadratic When Certain Points Are Known
The question is as follows:
Suppose $g(x)$
is a polynomial of degree $5$.
$g(0)=2$
and $g(1)=−623/15$
. Moreover, suppose $g(x)$
has local extreme values at $2√2$,$−2√2$,$√6$
and $√-6$
. Complete this ...
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answers
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Find the max and min of $(a \mathbf{x} + c)(b \mathbf{x} + d)$, where $\mathbf{x}$ is a vector with linear constraints.
Suppose there is a $n$-dimensional column vector $\mathbf{x}$, and an objective function $f(\mathbf{x}) = (a \mathbf{x} + c)(b \mathbf{x} + d)$, where $a$ and $b$ are $n$-dimensional row vector; $c$ ...
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To find the number of times the function takes the same value in the given interval.
The number of times the function $$f(x)=|\min(\sin x,\cos x)|$$
takes the value $0.8$ between $\frac{20 \pi}{3}$ and $\frac{43 \pi}{6}$ is :
$A)2,\quad$ $B) \text{more than}\, 2,\quad$ $C)0, \quad $ ...
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Irregular points for KKT theorem; $\min(x-4)^2+y^2$ for $x^2+y^2-9\le0,-x-3\le0,x-3\le0$ and $-y-2\le0$
I am solving this optimization problem:
\begin{equation}
\min (x-4)^2+y^2
\end{equation}
subject to
\begin{cases}
x^2+y^2-9\le0 \\
-x-3\le0\\
x-3\le0\\
-y-2\le0
\end{cases}
In the point (0,3) the ...
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2
answers
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Max $P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$
Let: $a,b,c>0$. Find the maximum value of:
$$P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$$
Here are my try:
I tried to use tangent line trick, then I got:
$$\...
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votes
1
answer
66
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Determine the extremum if any of $f(x,y) = 2x^4 - 3x^2y + y^2$.
My work.
$$f(x,y) = 2x^4 - 3x^2y + y^2.$$ We differentiate partially and find $$f_x = 0$$ and $$f_y=0,$$ which is indeed $0$ at $(0,0)$. Now $f_{xx} = 0$, $f_{yy}=2$ and $f_{xy}=0$. Since $f_{xx}*f_{...
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0
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25
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Squaring the function in maxima and minima
Whenever we have a square root in our function , we find it hard to differentiate hence we square the function and find the value of x such that the function is maximum or minimum.
for example: $$A=x\...
- 123
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7
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521
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Minimum value of expression $\displaystyle \sqrt{16b^4+(b-33)^2}$
Finding point $P(a,b)$ on parabola $x=4y^2$ whose distance from the point $Q(0,33)$ is minimum and also find that minimum distance
What I try : Let coordinate of point $P$ be $(4b^2,b)$ because point ...
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-4
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1
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56
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Maximum length of a side of an acute triangle with a fixed area of 1. [closed]
Is there a maximum length (or at least an upper bound) for a side of an ACUTE triangle with a fixed area? If there is , can it be found?
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2
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86
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Is there a name for functions that have multiple minima such that all minima are equal/one minima is less than another?
Is there a name for a concept like that I can search? I am completely okay if there is no name so far.
For example, $f\left(x\right)=\sin\left(x\right)$ has infinite minima, but they are all at $f\...
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votes
5
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What is $\lim\limits_{n\to\infty}\frac1n \left(\text{maximum value of }\sum\limits_{k=1}^n\sin (kx)\right)$?
Consider $f(x)=\sum\limits_{k=1}^n\sin (kx), 0\le x \le \pi$.
Here is the graph of $y=f(x)$ for $n=8$.
I noticed that, as $n\to\infty$, the maximum value of $\frac1n f(x)$ seems to approach a limit ...
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1
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Is minimizer containment preserved after adding the same function?
Suppose I have three convex real-valued functions $f$, $g$, and $h$, which satisfy $0\leq h\leq g$ and
$$\textrm{argmin}\, g\subset \textrm{argmin} \, h.$$
Is it true that $\textrm{argmin} (f+g)\...
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1
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41
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Differentiate expression involving an integral
How can I minimize this expression with respect to $c_i$ i.e. differentiate this expression, equate to zero and solve for $c_i$. The expression is $\sqrt{\int_a^b \left| f(x) - \sum_{i=1}^k c_i \phi_i(...
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0
answers
31
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Having doubt about proving whether product of increasing and decreasing function have minimum under some conditions.
Let, $f:(0,1]\to (0,\infty)$ be an increasing function with $\lim_{x\to 0} f(x) = 0$ and $g:(0,1]\to (0,\infty)$ be a decreasing function with $\lim_{x\to 0} g(x) = +\infty$. Also, if $\lim_{x\to 0} ...
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1
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31
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Minimising the upper bound of a constant
Let $C$ be a constant, $0\leq\gamma\leq1$ and $0<M<1$. Given that
$$
C \leq -\gamma x+\frac{x^2}{M-5x},\qquad \forall x\in[0,M/5).
$$
I want to give an upper bound of $C$ which is independent of ...
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2
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84
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Is there an alternative way to solve this trivial problem?
I dealing with a trivial problem:
Let $f(x) = xy$ be some area. Given $1500 = 15x + 6y$, find $x$ and $y$ so that the area is maximized.
Usually pretty easy when you can just plug in $(1500 - 15x)/6$...
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1
answer
49
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Let $M$ be the set $M = ( \frac{1 + 2n^2}{1 + n^2} \in \mathbb{R} : n \in \mathbb{N_0} ) \subset \mathbb{R}$. Find Inf, Sup, Min, Max
Let $M$ be the set
$$M = ( \frac{1 + 2n^2}{1 + n^2} \in \mathbb{R} : n \in \mathbb{N_0} ) \subset \mathbb{R}$$
Determine Infimum, Supremum, Minimum and Maximum of $M$, if those exist
Minimum: $n = 0$ ...
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2
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Notation for the "k" largest elements of array
Introduction. Given a set, or a vector, or a sequence $A$ with components:
\begin{equation}
A = \{a_1, a_2, a_3, \ldots, a_n \}
\end{equation}
we can indicate the maximum of $A$ as $\max(A)$, or as $\...
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1
answer
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Maxima and Minima of $x+\frac{1}{x}$
$$f(x)=x+\frac{1}{x}\\ \therefore f^{'}(x)=1-\frac{1}{x^2}\\
\text{now, } f^{'}(x)=0\ \Rightarrow x=+1,-1\\
f^{"}(x)=\frac{2}{x^3}\\f^{"}(1)>0\ \&\ f^{"}(-1)<0\\
\max f(x)=-2<\min f(x)=2$$...
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0
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Is it possible to analytically compute maximum/minimum within an area for a 2D function defined in an area?
For Easom function (or any other functions): $f(x,y)=\cos x*\cos y*\exp\left(-(〖x-π)〗^2-(y〖-π)〗^2 \right)$, defined in a range $0\le x,y\le 10$, the plot will look as follows:
enter image description ...
- 1