Mathematics Stack Exchange
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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.

3734 questions
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Sum of two max operators

Can the sum of two max operators be simplified and written as a single max operator? I have the following forms: $$\max(a-b,0) + \max(b-a,0)$$ and $$\max(a-b,0) + \max(c-d,0)$$
Basley
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How do I find the maximum value of $S(\tau) = (\tau^\eta − 2η\tau^{\eta−1} + \eta(\eta − 1)\tau^{\eta−2})e^{−\tau}$?

Given this function: $$ S(\tau) = (\tau^\eta − 2η\tau^{\eta−1} + \eta(\eta − 1)\tau^{\eta−2})e^{−\tau} $$ How can I find out what the maximum value of $S(\tau)$ is? I need this value in order to get a normalized form of this wave shape. I could…
Hamish McKenzie
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how to solve the minima?

Given x＞0，y＞0, 1/x ＋ 8/$y^2$ ＝1, solve the minima of $x$＋$y$? One guy has this problem solved by Cauchy inequality. Can anyone have another approach or? I worked out like this: if 1/$x$ ＝ 8/$y^2$ ＝ 1/2 hold, then $x$＝2, $y$＝4， (1/$x$)(8/$y^2$)＝1/4 ⥤…
Steve Young
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Find the maximum and the minimum value of x if $\frac{\sqrt{100-x^2}+\sqrt{99+x^2}}{40}= \cos \frac{\pi}{x^2-2|x|+4}$.

Find the maximum and the minimum value of $x$ if $$\frac{\sqrt{100-x^2}+\sqrt{99+x^2}}{40}= \cos \frac{\pi}{x^2-2|x|+4}$$ I just did $100-x^2\geq0\implies-10\leq x\leq10$, but what I have to do now? Is the answer -10 and 10?
Tas
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Maximum and minimum of $f(x)=\frac{1}{1+x^2}$

Is there a way how to calculate maximum and minumum of $f(x)=\frac{1}{1+x^2}$ without taking the derivative of it?
abcabcabc
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Find $\min(x+y)$ knowing that $x2^k+y=m$

Let $k$ and $m$ be specific numbers and $x,y$ such that $x(2^k)+y=m$. Find $\min(x+y)$
CodingNewbie
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Continuous and Differentiable Function has a Local Maximum

If $f$ is a differentiable function such that $f(a)=f(b)=0$, and there exists $c \in (a, b)$ such that $f(c)>0$, prove that $f$ has a positive local maximum.
minimario
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Maximum value of $x_1^2+x_2^2+\dots +x_k^2$

While trying to solve a problem in graph theory I need to find the maximum value of $x_1^2+x_2^2+\dots +x_k^2$ subject to a condition that $x_1+x_2+\dots + x_k=n$. Could someone guide me how to do this? Thanks in advance.
manifold
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For $x,y$. Minimize and Maximize $\left(x+y\right)^2-\sqrt{9-x-y}+\frac{1}{\sqrt{x+y}}$

For $x,y$ satisfy $x+y-1=\sqrt{2x-4}+\sqrt{y+1}$, minimize and maximize $$\left(x+y\right)^2-\sqrt{9-x-y}+\frac{1}{\sqrt{x+y}}$$
user413132
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Lots of local maxima-minima, limited solutions for critical points

I know how local & absolute maxima or minima are calculated, but I don't understand that if we differentiate a function once & put it equal to zero, we get $1,2$ or any other limited number of points of maxima or minima. But as we all know that…
shashank tyagi
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Maximizing revenue for ticket price

A rail company, QNRail, is selling tickets for a special trip from Toronto to Winnipeg. The train’s maximum capacity is 200 passengers. All tickets are sold at the same price p in Canadian dollars. Based on market research, they know that the…
jj889
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max $\sin(x)/x$ without derivative

Showing $max(\frac{sin(x)}{x})=1$ is straight forward using l'hopital's rule. Is there another way to evaluate without using l'hopital's rule
Christopher Mahoney
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Absolute max and min of $f(x, y) = x^{2} + 3y^{2} - y$ over the region $x^{2} + 2y^{2} \leq 1$

Find the absolute maximum and minimum values of the function $$f(x, y) = x^{2} + 3y^{2} - y$$ over the region $x^{2} + 2y^{2} \leq 1$
Chiranjeevi Star
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Equations maximize and minimize

Find the maximum and minimum of the equation: $x^2-x+\frac{1}{x^2}+x+1$. I am randomly trying to substitute values for $x$ but I need a complete method to solve such problems.
Mary Lopez
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Does this multivariable non-negative function only have minima points?

Consider multivariable functions of the form: $$\left(\sum_{i=1}^n a_ix_i\right)^2$$ Can they only have minima points? If so, why? I tried to plot some functions on Desmos and it looks like my hypothesis is correct, but maybe I'm missing…
Rony
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