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).
3,960 questions
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Minimizing $\sum\limits_{i=1}^n (x_i^3 - x_i^2)$ subject to $\sum\limits_{i=1}^n x_i = S$
The following is an algebraic problem I encountered in my research direction.
Let $x_1, \dots, x_n \ge 0 $ be non-negative real numbers satisfying $\sum\limits_{i=1}^n x_i = S,$ where $S \ge 0 $ is ...
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Find the minimum value of a symmetric function under the condition $(a+b)(b+c)(c+a)=1$ [closed]
I'm working on the following problem and would appreciate some guidance on how to approach it.
Problem.
Let $a, b, c > 0$ satisfy $(a+b)(b+c)(c+a) = 1$, and assume that $a, b, c$ are not all equal....
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How to prove that $f(\theta)$ reaches its minimum when $w_1w_2=0$?
I have the following function:
$
f(\theta)=\int_{-1}^{1} \max\!\left(0,\; \min\!\left(1,\; \frac{v - w_1 \xi}{w_2}\right) \;-\; \max\!\left(-1,\; \frac{-v - w_1 \xi}{w_2}\right)\right)\, d\xi,
$
where
...
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Relative and absolute extrema
I am having some doubts about the definition of relative and absolute extrema, which have arisen from the function $f(x)=\sqrt[3]{(x^2-1)^{2}}$.
For me, this function has:
Relative maximum but not ...
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Maximum value of $(a^2 + bc)(b^2 + ac)(c^2 + ab)$ when $(a+b)(b+c)(c+a) = 2$ for $a,b,c \in \mathbb{R_{\ge 0}}$
Let $a,b,c$ be non-negative real numbers satisfying $(a+b)(b+c)(c+a) = 2$.
Find the maximum value of the expression
$$P = (a^2 + bc)(b^2 + ac)(c^2 + ab).$$
Context:
I am training to get selected into ...
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Maximize eccentricity over all conics tangent to four fixed non-parallel lines
First, let's agree on the eccentricity of degenerate conics:
The animated gif shows Ellipses, hyperbolas with all possible eccentricities from zero to infinity and a parabola on one cubic surface. ...
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How do transformations by exponential and logarithmic functions affect monotonicity and extrema of a sequence?
Let $f(n)$ be a real-valued sequence defined for $n \in \mathbb{N}$, with $f(n) > 0$ for all $n$.
Define a new sequence:
$$
g(n) = \log_b(f(n))
$$
I know that when $0 < b < 1$, the ...
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How to find the minimum value of $ f(x)=\frac{(e^{x}+x)(1-xe^{x})}{(e^{2x}+1)(x^{2}+1)}$
I would appreciate if somebody could help me with the following problem
For all real numbers, consider the function
$$
f(x)=\frac{(e^{x}+x)(1-xe^{x})}{(e^{2x}+1)(x^{2}+1)}.
$$
Find its minimum value. ...
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If a function has a discontinuity point, are extrema still possible at that specific point? [closed]
I have been looking into it for some time now and can’t seem to digest it. Suppose there exists a function $f(x) = x^2$ where its global minimum is at $x = 0$ and $f(x) = 0$. However, there is a hole ...
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Find the minimal value of $a^2+b^2+c^2+d^2$ given that $a+b+c+d=p>0$ and $abcd=1$
Problem. Let $a,b,c,d$ be real numbers such that $abcd=1$. What is the minimal
value of $P=a^2+b^2+c^2+d^2$ under the following conditions:
a) $a+b+c+d=3$
b) $a+b+c+d=\frac{49}{8}$
My attempt is not ...
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Power-sum of areas of inscribed triangles from 4 cyclic points: sharp bound
Question.
Let $P_1, P_2, P_3, P_4$ be four distinct points on the unit circle in that cyclic order, and let $S_{ijk}>0$ be the areas of the inscribed triangles $\triangle P_iP_jP_k$. For $p\ge2$ ...
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Greatest-eccentricity circumconic of a concave quadrilateral
Nine-point conic $\cal N$: the unique conic through the midpoints of the $\binom42=6$ segments joining the $4$ vertices.
Gergonne–Steiner conic $\cal G$: the unique conic in the pencil of conics ...
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Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$. If P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum, find $a$ and $b$
Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$.
If point P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum,
then find the value of $a$ and $b$.
My Attempt:
If $P$ lies ...
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Find the maximum value of $\frac{ab}{2a+b} + \frac{3bc}{2b+c} + \frac{6ca}{2c+a} $
Let $a,b,c>0$ satisfy $13a + 5b +12c=9$. I want to find the maximum value of
\begin{align}
A = \dfrac{ab}{2a+b} + \dfrac{3bc}{2b+c} + \dfrac{6ca}{2c+a}.
\end{align}
I tried applying AM–GM like so :...
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Find maximum of $3^ba^{b+1}$ for positives $a+3^b=6$ using derivatives
The following problem is from a contest in Yunnan, China.
If positive real numbers $a$ and $b$ satisfy $a+3^b=6$, find the maximum of $3^ba^{b+1}$.
Here is one solution: let $a=3^c$, then by AM-GM,
$...
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SOLVED: Maximum semi-major axis of an ellipse of 2pi circumference - the gamma function?
Edit: I probably just got over my head and my approximations were off, resulting in a coincidence.
I was doing some numerical exploration with ellipses as a layperson. It occurred to me that if you ...
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Showing a relationship between cauchy schwarz inequality and geometrical area
I was reading a textbook and I couldn't quite understand one part of the solution. It stated, that the vertices of a fixed triangle are points $A$, $B$ and $C$. Then points $P$, $Q$ and $R$ lie on the ...
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Find $k$ such that $(ab+bc+ca-1)^2\ge k\cdot abc(a+b+c-3)$
Problem. Find best constant $k$ in which:
$$(ab+bc+ca-1)^2\ge k\cdot abc(a+b+c-3)$$ holds for all $a,b,c\in R: ab+bc+ca+1=a+b+c$
I have a proof for $k=2$ but I am not sure that $k=2$ is the desired ...
user1656827
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Geometric approach to find the triangle of smallest perimeter which circumscribes a semicircle
The problem to find the triangle of smallest perimeter which circumscribes a semicircle, was solved by DeTemple here, using derivatives.
I wonder if a purely geometric solution can be found for this ...
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Find radius of ball causing the greatest volume to overflow
A Martini glass in the shape of a right-circular cone of height $h$ and a semi-vertical angle $\alpha$ is filled with liquid.
Slowly a ball is lowered into the glass, displacing liquid and causing it ...
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How to find the maximum of a difference of products?
Let $$f(x) := \prod_{i=1}^n (a x_i + b) - \prod_{i=1}^n (c x_i + d),$$ where $a,b,c,d$ are positive constants, and $x$ is a real vector of length $n$.
I want to find $\max f(x)$ subject to $0 \leq x_i ...
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Minimize $a^2+b^2+c^2+d^2$ subject to three quadratic conditions [duplicate]
Let real numbers $a$, $b$, $c$ satisfy $\begin{cases}(a+b)(c+d)=25,\\(a+c)(b+d)=20,\\(a+d)(b+c)=7\end{cases}$, find the minimum of $a^2+b^2+c^2+d^2$.
This is the last problem in the O level exam of ...
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Minimum of $T=4\sum_{i=1}^nx_i^3+3(1-\sum_{i=1}^nx_i^2)^2$
Let $n\geq2$ be a natural number and $n$ nonnegative real numbers $x_1,x_2,...,x_n$ such that $\sum_{i=1}^nx_i=1$. What is the minimum of $T=4\sum_{i=1}^nx_i^3+3(1-\sum_{i=1}^nx_i^2)^2$?
Take a ...
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Minimum of the density of a wrapped normal distribution
Let $x_0\in[0,1)^d$, $\Sigma\in\mathbb R^{d\times d}$ be positive and symmetric, $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor$$ (floor applied componentwisely) and $$\pi:=\mathcal ...
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When does convergence of a function at a sequence of points to its max imply points convergence to the argmax?
Consider a function $f$ over an Euclidean domain $\mathbb{X} \subset \mathbb{R}^d$, for some $d \in \mathbb{N} \setminus \{0\}$. Assume the function is bounded and that the set $\mathbb{X}^*=\arg\max_{...
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