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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.

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For $a,b,c>0$. Minimize $P=a+b+c$

For $a,b,c>0$ and $\frac{2}{a}+\frac{5}{b}+\frac{3}{c}=1$, minimize $$P=a+b+c$$
Word Shallow
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Maximum value of $f(x, y, z) = e^{xyz}$ in the domain x+y+z = 3.

Need the maximum value of f$(x, y, z) = e^{xyz}$ in the domain $x+y+z = 3$. Please help me on this, as I could not find how to solve this one. Thanks in advance.
Sivabalanarayanan L
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The value of parameter a for which $\frac{ax^2+3x−4}{a+3x−4x^2}$ takes all real values for $x\in \mathbb R$ are:

I solve it and get the answer $a\in [1,7]$.but my teacher told me to take the verification of the boundary values of a.because at the boundary values, $ax^2+3x-4$ and $3x-4x^2+a$ have common roots.so,its obvious that value of a will not count in the…
soayel akter
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Did I spot a mistake in Riley, Hobson, Bence?

Let $f({\bf x})\equiv f(x_1,x_2,\ldots,x_n)$ is a function of $n$ real variables ${\bf x}=(x_1,x_2,\ldots,x_n)$. The Taylor expansion of $f({\bf x})$ is about a local ${\bf x}^0=(x_1^0,x_2^0,\ldots,x_n^0)$, reads, $$f({\bf x})=f({\bf…
SRS
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Maximum area of triangle, tangent

In the following problem: Consider the set of all triangles whose sides are the x and y axes and tangents to the curve e^{-4 x}, x>0. Calculate the maximum area such a triangle can have. I have tried to solve it, but the answer is apparently wrong,…
Kei Len
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Find the maximum of a function

Function given: $$f(\alpha, b,\delta) = \left(a_0 + a_2\cdot(\alpha-\delta)^2 + a_4\cdot(\alpha-\delta)^4\right)\cdot\left(1-\frac{b-1}{2b}\right)\cdot e^{j\cdot \pi} + \left(a_0 + a_2\cdot(\alpha+\delta)^2 +…
Vaisala
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Finding where a function is increasing or decreasing: regions to check

I'm solving a problem where I need to determine where a function is increasing and decreasing. I know that I need to check the regions between the critical points. However, the proposed solution to the problem also says that I need to include points…
Mathematical Endeavors
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Do shapes with maximum area also have maximum perimeter, and solids with maximum volume also ahve maximum surface area?

I just want to ask if all shapes that have maximum area also have maximum perimeter and if all solids that have maximum volume also have maximum surface area. Why is this so? I just find it interesting that when it comes to shapes and solids, two…
AndroidV11
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Maximum of nothing

Say I have some function where a certain condition cannot be fulfilled for certain inputs, e.g. $$f(x)=\max_{a\in \mathbb{N}, a < x}\quad a^2$$ and I plug in $x=-3$. Then there is no number $a$ such that $a\in \mathbb{N}$ and $a<-3$, hence I cannot…
Duck71
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Maximize $(z-x)$ such that $x^2 + y^2+ z^2 =1$

Here I tried the coordinate geometry, as in the equation represents a sphere. From there $(z-x)$ would imply the distance between the $z$ coordinate and $x$ coordinate so that the difference is maximum. Since it is a sphere with radius $1$, $z-x =$…
Ramesh Sharma
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a minimimization problem involving a ladder across a fence, to a building; the twist- no constants are given; Need help!!

From Calculus, Varberg and Purcell, 6th Edition; am teaching myself from the text; no answer or explanation is given in the text. I believe I was able to successfully obtain the correct (minimized) derivative for the value of a new variable I…
Victor Jaroslaw
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How can I decompose the $\max$ of a product of functions?

In this video, the author states that if $f(x) \geq 0$ for every $x$ and $g(x,y) \geq 0$ for every $x$ and $y$, then $$ \max_{x,y} f(x)g(x,y) = \max_x \left[f(x) \cdot \max_y g(x,y)\right] $$ I tried to prove this equality using $\log$'s as…
mhdadk
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Can relationship between two variables in maxima and minima be solved using partial derivatives?

In obtaining the maxima and minima of the volume of solids, it is common to express the equation in terms of one variable and then apply derivatives. Like for example, a rectangular prism with volume $ V = x^2 y $ and $ SA = 2x^2 + 4xy $ has its…
AndroidV11
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Help with maxima and inflection points, why are my answers incorrect?

Find all the maxima and inflection points of the following function: $$f(x)=\frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}}$$ Answer is local Max is at $(0,\frac{1}{\sqrt{2\pi}})$ inflection points $(1,\frac{1}{\sqrt{2\pi{e}}})$ &…
MCO
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