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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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Prove that minimum of an averaged array is equal to an average of minimums of original arrays

During coding specific algorithms I required an averaged of minimums of three arrays (they are same size), but I have only access to averaged array of three mentioned above. During testing numbers appears to be the same, but I have no luck…
Digoya
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Minimum value of Sum for reals

Find minimum value of Sum $\displaystyle\sum _{k=1}^n a_k^2$ for reals satisfying $\displaystyle\sum _{k=1}^n a_k=1$ . This is first time I am encountering such problem . How to approach such problem where there are so many variables and a little…
RKK
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find maximum value and minimum value

$y=x^2$ and $D_f=(0,2]$. So, my interval is $0
user876873
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Maximum $m:=\max${$u(x)$ s.t $x \in A\bigcup B$}

How can we express $m$ denoted by $m:=\max${$u(x)$ s.t $x \in A\bigcup B$} as comparing it with $\max${$u(x)$ s.t $x \in A$} and $\max${$u(x)$ s.t $x \in B$} What about minimum? I need only clarification, you can without proof if you have no time.
sara
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How to find the maximum value of this expression?

expression:$$\sum_{i=1}^{n-1} \frac{\sqrt{1+\cos(\theta_i)}}{\sqrt{2}} +\frac{\sqrt{1-\cos(\sum_{i=1}^{n-1}\theta_i)}}{\sqrt{2}}$$ I guess its maximum value is obtained when all $\theta_i=\pi/n$ are the same and the the max value is…
Zhou
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Minimum ticket price to break even

A football team plays in a stadium that has a seating capacity of $30,000$ spectators. With the ticket price set at $\$79$, average attendance has been $21,692$. A market survey indicates that for each $\$4$ the ticket price is lowered, the average…
Gulnoor
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Solving for x in a specific range

I am trying to find the minimum value, analytically, of a somewhat complicated function, and I've been struggling with it all day. It's complicated (to me) as the function is a sum of a periodic function (in $x$) and a function that is not…
Jason Rohr
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Investigate local maxima and minima

Question: $x^2 + 4xy + 4y^2 + x^3 +2x^2y+y^4 $ critical point $(0,0)$, $F_{xx}*F_{yy} - (F_{xy})^2$ = $0$ now test failed to conclude any thing. But solution says it has minimum at the origin. Please guide me how to proceed. And lastly, I am…
Abhishek Verma
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$x^2 - y^2=9$, $2y-x=3$, Find maximum value of $\dfrac{y}{x}$

I came across this question and have no idea what to do. Given, $$ x^2-y^2=9 \\ 2y-x=3 $$ find maximum value of $\dfrac{y}{x}$
seraphimk
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Minimise the function $f(x) = \frac{x^2 - x +4}{x-1}$ using Calculus

I had to minimise the function $$f(x) = \frac{x^2 - x +4}{x-1}$$ I did the method where I found the range of this function and found the minimum value. However I know some basic calculus and was trying to find it using that but I am not able to. So,…
Rasputin
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Finding the extreme value of a function

Let f(x) be a continuous function such that $f’(x) = (12x – 48)/[3(x – 4)^2 + 1]$ for all real numbers x. (a) If f(x) attains its minimum value at x = k, find k. (b) It is given that the extreme value of f(x) = 5. Find f(x) and lim f(x) when x tends…
Mick
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Extrema of a function

Consider the function $f(x,y) = -10x^2-10xy+30x$ I am about to find the extrema or saddle of the function. Calculated critical points, $f_x = 0$ $-20x-10y+30$ $f_y = 0$ $x=0$ $y=3$ Critical point is (0,3) $A = f_{xx}(0,3) = -20$ $B= f_{xy}(0,3) =…
Aruha
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If two positive integers x and y such that $x + 2y = 60$ then find the max vaue of xy?

Can anyone please solve the question with steps I did this by just assuming that let give both $x$ and $2y$ equal value $30$, $30$ so $x\times y = 30\times15 = 450$ (By the way $450$ is the correct answer) Can anyone do this problem in steps.
Vaibhav Rawat
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Minimize function - nested square roots

I would like to find $z$ which minimizes the below, when $x$ is held at a specific value. $f(x,z) =\sqrt{\sqrt{x^2 + z^2} - 0.25}$ For example; I would like to find the value of $z$ which minimizes the function when $x = 0.5$
brent
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Find Minima for the equation

Minimum point for $x^2 +2xy + 4y^2 + 6$ is to be found. $\frac{\partial}{\partial x}$ and equated to $0$ to find $x=-y$ $\frac{\partial}{\partial y}$ and equated to $0$ to find $x = -4y$ I'm confused as to which is the correct solution
Orpheus
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