Mathematics Stack Exchange
Mathematics Stack Exchange

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
1
vote
2 answers

circle cut into twelve equal-area rectangles

I want to cut circle into twelve equal-area rectangles 【Use the methods in the picture】(the length and width of each rectangle can be different). To make the area of ​​the rectangle is maximum, how to cut it? It is allowed for a rectangle of the…
kapike
  • 121
1
vote
1 answer

Maxima of increasing function multiplied by its symmetryc

Let $f$ be a function with the following properties: $f$ is strictly increasing $f(0) = 0$ $f(1) = 1$ What are the maxima of $g(x) = f(x)f(1 - x)$? More specifically: Under which conditions do we have that $x=\frac{1}{2}$ is the only maximum for…
Matteo Monti
  • 341
  • 2
  • 9
1
vote
1 answer

If $f(x)^2$ is convex and $f(x)>1$, does $\arg\min_x f(x)^2=\arg\min_x f(x)$?

Not sure, maybe it's trivial... Thought about that during my shower this morning. My intuition is as follows. Let $x'=\arg\min_x f(x)^2$. For all $h$, we should have $$f(x+h)^2\ge f(x)^2$$ Then, since $f(x)>1$, we have $f(x)^2\ge f(x)$. Therefore,…
davcha
  • 1,745
1
vote
4 answers

Suppose $x, y, z$ are positive real number such that $x + 2y + 3z = 1$. Find the maximum value of $xyz^2$

I'm trying to solve this problem using Lagrange's multiplier method but I'm unable to get the value of lambda and also $z.$ kindly help me out with this problem. I'm getting $x = 2y$ I'm stuck somewhere between the steps
Priya Wadhwa
  • 173
1
vote
1 answer

How do you find the maximum of a set like {$\frac{2n+(-1)^n}{n+2}, n \in \mathbb N$}?

I don't know how to go about finding maxima/minima of such sets. What about a set like {$\frac{m+n}{m+2n}, n,m \in \mathbb N$}? I'd like to treat the numerator and denominator seperately, and then say that the ratio of the max over the min is the…
iaskdumbstuff
  • 815
  • 1
  • 6
  • 14
1
vote
1 answer

Use maxima to find $x$ and $y$ coordinates of the local maximum

Ok, so I have function $$f(x)=\dfrac{x^3+6x^2-3x}{x^2+1}$$ All answers have to be completed on maxima. The first part is to plot the graph, which I have. The second part is to find the derivative of $f$. I have also completed that. Part C asks…
Dogmama
  • 21
1
vote
1 answer

1st and 2nd derivative test of max-min

Is there any situation where 1st derivative test is preferred over 2nd derivative test or 2nd derivative test preferred over 1st derivative test? Or do they prefer equivalently? I have been running across so many problem where the two test can…
demon
  • 125
1
vote
1 answer

Cauchy-Schwarz? AM-GM? Minimum of function

$x\ge -\dfrac{1}{2}; \dfrac{x}{y}>1$ Find the minimum of the function $\frac{2x^{3}+1}{4y\left ( x-y \right )}$ I thought of using a few values of $x$ and $y$, but that seemed extremely inefficient. I've seen many people describing AM-GM and…
A Piercing Arrow
  • 953
1
vote
3 answers

Find the minimum value of the function in 2 variables.

Find the minimum of the function of two variable defined as $$f(x_1,x_2)=(x_1-x_2)^2 +\left(2\sqrt2x_1-\sqrt{6(x_2)-(x_2)^2-7}\right)^2.$$ At first I tried to figure out a integral in this but failed. Then I tried using graphs, but things became…
user354545
  • 577
1
vote
2 answers

Prove area of triangle with a given base and a given vertex angle is maximum when triangle is isoceles

Consider the set of triangles having a given base and a given vertex angle. Show that the triangle having the maximum area will be isosceles. I have taken $a$ and $\alpha$ to be the given base and vertex angle respectively. Area of triangle in terms…
user467745
  • 604
1
vote
1 answer

Minimise this function?

I'd like to minimise a function like: $$ R^2(x) = \sum_{i=1}^N (Y_i - \frac {a_i}{x \cdot c_i+b_i})^2 $$ This is similar to Ordinary Least Squares, but seems to be harder to solve because the $x$ is in the denominator. Can this be solved?
user3766223
  • 169
1
vote
1 answer

Maxima and Minima problems

I'm having trouble with the following question about maxima and minima. I have to find the points of local extrema of the function, $$f(x,y)= 12x^2y+3y^3-48x^2 - \frac{81}{2}y^2+72y-4.$$ So, I begin with $$f'(x)=24xy - 96x $$ $$f'(y)=12x^2 + 9y…
Lam Bui
  • 11
1
vote
1 answer

Find the absolute minimum and absolute maximum

Find the absolute minimum and absolute maximum in [-1,2] of the function $ \ f(x)=|x| \ $. $$ $$ I have got absolute minimum at x=0 , ie., $ f(0)=0 $ is the absolute mimimum. The absolute maximum is $ f(2)=2 $ . Is this correct ?
MAS
  • 10,638
1
vote
2 answers

Find the minimum value of $\frac{x^2}{x-1}$ for $x > 1.$

Find the minimum value of $$\frac{x^2}{x-1}$$ for $x > 1.$ I can't use calculus, and I think the question is meant to be solved using the Trivial Inequality, the Mean Chain, and/or the Cauchy-Shwarz Inequality. Any help would be greatly…
Quantum Pizza
  • 146
1
vote
3 answers

The real number x when added to its inverse gives the minimum value of the sum at x equal to

The real number x when added to its inverse gives the minimum value of the sum at x equal to what? According to me it is 2 as $x +(1/x) $ is always equal to greater than 2. But the answer is given as 1.
a25bedc5-3d09-41b8-82fb-ea6c353d75ae
  • 1,769
1 2 3
…
9 10