# 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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### 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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### 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 ...
1 vote
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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 $$(3x+4y+7z)(1/x+1/y+1/z)\ge (\sqrt{3}+2+\sqrt{7})^2$$...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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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