What is Minimum: Definition and 1000 Discussions

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 (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
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.

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  1. M

    Shoot a basketball with a minimum speed at some angle

    Homework Statement You should shoot a basketball at the angle ##\theta## requiring minimum speed. Avoid line drives and rainbows. Shooting from (0, 0) with the basket at (a, b), minimize ##f(\theta)= 1/(a \sin (\theta) \cos (\theta) -b \cos^2 (\theta))##. (a) If b =0 you are level with the...
  2. R

    When is the minimum polynomial of a scalar matrix kI equal to t-k?

    Homework Statement Show that A is a scalar matrix kI if and only if the minimum polynomial of A is m(t) = t-k Homework EquationsThe Attempt at a Solution f(A) is monic f(A) = 0 since A = kI Next we must show that deg(g) < deg(f) I guess I'm not sure where g comes from. Is it merely an...
  3. synMehdi

    A Pontryagin minimum principle with control constraints

    Hi, I am trying to solve a control problem where I have to minimize the fuel consumption of a vehicle: $$J=\int_{0}^{T} L(x(t), u(t),t) + g(x(T),T)dt$$ ##L(u(t),v(t))=\sum\limits_{i,j=0}^{2} K_{i,j} u(t)^i v(t)^j ## is convex (quadratic) and the term ##g(x(T),T)## is to have a constraint in the...
  4. Physics345

    Minimum cost of an area of fencing using derivatives

    Homework Statement A homeowner wishes to enclose a rectangular garden with fencing. The garden will be adjacent to his neighbour’s lot. There will be fencing on all four sides. His neighbour will be paying for half the shared fence. a) What should the dimensions of the garden be if the area is...
  5. C

    Proving Snell's law using Euler-Lagrange equations

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  6. S

    Minimum force to separate two oppositely-charged ions

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  7. T

    Finding minimum and maximum intensity of waves

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  8. bbbl67

    I What is theoretical minimum mass of neutronium?

    Not withstanding the fact that most neutronium is produced in neutron stars, what is the lowest mass required to keep neutronium stable? I've heard it could be as low as 0.1 solar masses. Could it be even smaller than that? Can you make an Earth mass neutronium ball and have it stay stable due...
  9. Alexanddros81

    Calculate the maximum and minimum distances between the Earth and the Moon

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  10. Sheldon11

    B Why do we have minimum deviation of a prism?

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  11. V

    Minimum and maximum resultant of three vectors.

    Homework Statement In order to solve a question, I need to find the minimum and maximum resultant of three vectors, their magnitudes are given to me. Homework Equations Magnitude of vector A = 1 Magnitude of vector B = 3 Magnitude of vector A = 5 The Attempt at a Solution The maximum part was...
  12. K

    Calculate the minimum uncertainty

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  13. Peter Alexander

    How to determine minimum amount of AND and OR gates in PLA?

    Homework Statement Realization of given three functions $$ \begin{array}{c} f_{1}=x_{1}\bar{x}_{2}\bar{x}_{3}+x_{2}x_{3}\bar{x}_{4}\\ f_{2}=\bar{x}_{3}x_{4}+x_{1}\bar{x}_{2}\bar{x}_{3}\\ f_{3}=x_{2}x_{3}\bar{x}_{4}+\bar{x}_{3}x_{4} \end{array} $$ using the PLA. Homework Equations I don't know...
  14. G

    DNA testing, what kind of sample is the minimum?

    Is it possible to do DNA test from fallen hair, or skin cells? Or no, since the cells die and lose their core before they fall from the body? Do dead cells still have somekind of unique "fingerprint"?
  15. J

    DC Distribution System -- Minimum Voltage across the line

    Homework Statement Homework Equations V = IR And R becomes 2(r of one conductor) because of return path. The Attempt at a Solution V minimum is at third junction. At VD Using KVL we get: [/B] Drop at AB = (10 + 20 + 50) * 2 * 0.1 = 16 V Drop at BC = (20 + 50) * 2 * 0.08 = 11.2 Drop at CD =...
  16. V

    Minimum friction to roll an imperfect sphere on an incline

    Homework Statement I have a sphere, it's imperfectly spherical, I put it on an incline and apparently it needs a minimum friction to start rolling or moving. Homework Equations I = 2/5 MR^2 torque = I*alpha = R x F The Attempt at a Solution [/B] before anything, I don't get why there's a...
  17. bbbl67

    I What is the minimum mass of a neutron star?

    We just discovered the maximum mass of a neutron star, discovered after the recent neutron star merger event back in Aug. They say that the maximum mass of a neutron star is approximately 2.16 solar masses. So I always assumed that the lowest mass for one is 1.4 solar masses, the Chandresekhar...
  18. Roger Dodger

    B Least / Smallest / Minimum Detectable Difference

    In scientific experiment, we often have a physical property that can change but have no detectable impact on the measurement. For example, suppose I have a mass of (say) 30 grams attached to a string passing over a pulley. I can add up to another 2 grams and the system doesn't budge. In our...
  19. M

    MHB How Can We Find the Minimum 2-Norm of Ax-y Using an Orthogonal Matrix?

    Hey! :o Let $A = QR$, where $Q$ is an orthogonal ($m\times m$)−matrix and $R$ is an upper ($m\times n$)-triangular matrix of rang $n$ ($m>n$). I want to show that $$\min_{x\in \mathbb{R}^n}\|Ax-y\|_2=\|(Q^Ty)_{n+1}^m\|_2, \ \forall y\in \mathbb{R}^m$$ It is $(a)_k^l=(a_k, \ldots , a_l)^T$...
  20. Richie Smash

    Inequality to represent minimum monthly income.

    Homework Statement I Type I Cost per pupil I I Full session I $50 I I Half session I $30 I The above table shows the cost of lessons per month to students attending a private class. The class operates under the following limitations...
  21. BiGyElLoWhAt

    I Minimum number of moves from a to b

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  22. X

    Maximum and minimum tensile/compressive strain in beam

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  23. Abhimessi10

    Finding the minimum time for crossing a river

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  24. Pushoam

    Minimum consumption rate of a heat pump

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  25. kandelabr

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  26. EastWindBreaks

    Minimum number of teeth given the gear train ratio?

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  27. T

    Shear force for minimum links is greater than the maximum sh

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  28. T

    Calculate minimum work input of refrigeration cycle

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  29. Y

    Find minimum stopping distance

    Homework Statement A crate is being transported on a flatbed truck. The coefficient of static friction between the crate and the bed is 0.470. What is the minimum stopping distance if the truck, traveling at 70 km/h, is to decelerate uniformly and the crate is not to slide forward on the bed...
  30. K

    What is the Proof for Local Minimum and Maximum at First Derivative Equals Zero?

    Homework Statement Homework Equations At local minimum or maximum the first derivative equals zero The Attempt at a Solution a) $$f'=2x-\frac{a}{x^2},~~2\cdot 2-\frac{a}{4}=0~\rightarrow a=16$$ Near 0 from the left a/x gets large negative values, smaller than for a=16. that's my proof for...
  31. A

    Minimum spring force required to keep a car level

    Homework Statement This is an attempt to solve a problem I asked about here https://www.physicsforums.com/threads/can-a-car-ever-sit-like-this.929453/ Homework Equations F = kx spring force The Attempt at a Solution using the largest angle down it can get with the front spring compressed all...
  32. Oannes

    Finding Minimum Force Needed to Hold Object In Place

    Homework Statement A woman holds a book by placing it between her hands such that she presses at right angles to the front and back covers. The book has a mass of m = 0.75kg and the coefficient of static friction between her hand and the book is μs = 0.59. What is the minimum force she must...
  33. Muthumanimaran

    Minimum work needed to compensate for a heat leak

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  34. Noisy Rhysling

    Minimum population needed to keep an anthropoid creature....

    ...from suffering inbreeding and decline into a non-viable number of animals? Assume a creature somewhat larger than the average human. And with bigger feet. Mods, if this needs moved go for it.
  35. T

    Maximum entropy principle from minimum energy principle

    Homework Statement Formulate a proof that the energy minimum principle implies the entropy maximum principle. That is, show that if the entropy were not maximum at constant energy then the enrgy could not be minimum at constant entropy. HINT: First show that the permissible increase in entropy...
  36. M

    MHB Negative minimum of the average variable cost function

    Hey! :o Can the minimum of the average variable cost function be negative? (Wondering) Suppose we have the cost function $K(x)=x^3-9x^2+11x+100$. The variable cost function is then $K_v(x)=x^3-9x^2+11x$. The average variable cost is $k_v(x)=\frac{K_v(x)}{x}=\frac{x^3-9x^2+11x}{x}=x^2-9x+11$...
  37. M

    What is the minimum speed required for Spiderman to catch a falling baby?

    Homework Statement A baby falls from the top of a ##45 m## building. What is the minimum speed that Spiderman must push off from the same building one second later in order to catch the baby before she strikes the ground? Homework Equations Please see below. The Attempt at a Solution I tried...
  38. Pushoam

    Minimum acceleration of a car at which front wheels leave the ground

    Homework Statement Homework EquationsThe Attempt at a Solution I assume that the wheels are 8ft apart in both x and y direction. Let's denote the length and width of the car by 2l = 8 ft. The torque about an axis connected the contact points of rear wheels : ##\vec \tau = [\{l( \hat x +...
  39. B

    What is the minimum number of H+ ions required during ATP production?

    how many minimum number of H+ ions are required during ATP production in citric acid cycle (CAC)?
  40. lfdahl

    MHB Find the minimum a^4+b^4+c^4−3abc

    Find the minimum of the expression: $$a^4+b^4+c^4-3abc$$ - if $a,b$ and $c$ are real numbers satisfying the conditions: $a \ge 1$ and $a+b+c = 0$. Hint:
  41. I

    Minimum Mass required to lift up an object from a spring

    Homework Statement FInd min mass so that block loses contact with ground. Spring is initially in its natural length, the mass attached to spring is on ground(M) Homework Equations Equations of conservation of energy The Attempt at a Solution Min compression = 2mg/k to lift up object, Normal...
  42. A

    Minimum h for Car to Remain on Track Around Loop

    Homework Statement A car of mass m is initially stationary on a smooth track at distance h above the ground. What would be the minimum value of h required in order for the car to remain on the track throughout its journey around the loop of radius r? Homework Equations KE(bottom)>PE (top)...
  43. A

    I Max and min values -- Finding them in a data set

    I have some data collected from experiment and I want to calculate the max and min values for this data. Please note that these data not a result of function. What is the best mathematical way to get the max and min without just picking them by directly observation.
  44. J

    Minimum value of coefficient of friction

    Homework Statement Ans: c) Homework EquationsThe Attempt at a Solution I can solve this problem if only I understand the setup . I am sure the diagram is missing something . Can someone help me understand the problem statement in context of the setup ?
  45. Const@ntine

    Wave Interference: Minimum Intensity/Hyperbolic

    Homework Statement Two identical speakers, 10.0 m apart from each other, are stimulated by the same oscillator, with a frequency f, of 21.5 Hz, at a place where the speed of sound is 344 m/s. a) Show that a receiver at A will receive the minimum intensity of sound (Amin) due to the...
  46. |Glitch|

    B What is the relationship between brown dwarfs and giant gas planets?

    It has been my understanding that a brown dwarf was defined by when an object has achieved sufficient mass to begin deuterium fusion. Much like a star is defined by when it has achieved sufficient mass to begin hydrogen fusion. The minimum mass to begin fusing deuterium is estimated to be...
  47. K

    B Calculus of variation. Minimum surface

    so df/dy' is yy'/ √(1+y'^2) then we are supposed to do y' . [ yy'/ √(1+y'^2) ] - y√(1+y'^2) how does this bring equation 2 in the image ?
  48. E

    Vertical Force Value Needed

    Homework Statement A block of mass 7 kg is pushed against a wall by force P. The coefficient of static friction between the block and the wall is 0.42. Determine an expression for the force P as a function of θ such that the block will not slide up the wall or fall down the wall. a.) Draw a...
  49. T

    Arranging resistors to have a minimum equivalent

    Homework Statement The minimum resistance obtained by connecting 50 resistances of 0.25 ohm each is A)50/4ohm B)4/50 ohm C)200 ohm D)1/200 ohm 2.Relevant equations 1.When resistances are connected in series, ▪R(equivalent)=summation of all individual resistances 2.When resistances are...
  50. D

    Could a distribution of magnetic Dipoles create an Energy minimum

    The think is that I would like to create a magnetic trap "minimum of magnetic potential energy in all 3 directions of space" using a distribution of magnetic dipoles as a field generator. In other words, I would like to know if setting in some way some magnetic dipoles is possible to create a...
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