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. evinda

    MHB Find Min Element in A Sequence w/ $O(\log n)$ Complexity

    Hello! (Nerd) A sequence of numbers $A=(a_1, a_2, \dots, a_n)$ is called convex if there is a $k$ with $1 \leq k \leq n$ such that $\forall 1 \leq i \leq k-1$, we have that $a_i>a_{i+1}$ and $\forall k \leq i \leq n-1$ we have that $a_i<a_{i+1}$. Write an algorithm with time complexity $O(\log...
  2. R

    Finding the minimum force F which must be exerted

    Homework Statement Given: The friction between the block with mass 6 kg and the wedge with mass 18 kg is 0.27 . The surface between the wedge with mass 18 kg and the horizontal plane is smooth (without friction). The acceleration of gravity is 9.8 m/s^ 2. A block is released on the inclined...
  3. HaLAA

    How Fast Must the Boat Travel to Clear the Shark Tank Stunt?

    Homework Statement You've taken a summer job at a water park. In one stunt, a water skier is going to glide up the 2.0-m-high frictionless ramp shown, then sail over a 5.0-m-wide tank filled with hungry sharks. You will be driving the boat that pulls her to the ramp. She'll drop the tow rope at...
  4. M

    Piezoelectric and minimum dimensions of an actuator beam?

    Homework Statement We want to build a micro-positioning system that produces a maximum travel of 5 μm by using a piezoelectric linear actuator beam with a square cross section (a^2) and length (l) as shown below. A piezoelectric material with piezoelectric coefficients d13 = -2.0×10-8, d33...
  5. U

    Minimum numbers of balls to be drawn

    Homework Statement suppose in a box there are 20 red, 30 black,40 blue,and 50 white balls.what is the minimum numbers of balls to be drawn, without replacement,so that you are certain about getting 4 red, 5 black, 6 blue and 7 white balls? Homework EquationsThe Attempt at a Solution so there...
  6. L

    Finding minimum push force for a table

    Homework Statement The coefficient of static friction between a static friction and a 20kg table and the floor is 0.29. The coefficient of kinetic friction between the same surfaces is 0.25. How much minimum push force is required to get the table moving? How much push force is required to...
  7. J

    How Fast Must a Ball Swing to Complete a Circle?

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  8. C

    Finding the absolute minimum of a function in R2

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  9. D

    What are the minimum and maximum temperatures?

    Homework Statement A 1-kg steel rod is at 227C and is dropped into a 1-litre bath of water at 27C. Assume that these materials are thermally insulated from the rest of the universe. You will determine the final temperature of the steel rod after it reaches thermal equilibrium with the water...
  10. S

    What is the minimum frequency to keep the mass moving?

    Homework Statement A mass of 2.0 x 10^2 g is tied to a 1.6 m long string and spun in a vertical circle. What is the minimum frequency to keep the mass moving? Homework Equations Fc=m4(pie)rf^2 The Attempt at a Solution I know I have to use this equation and make it equal to another to find Fc...
  11. A

    Help Determining Minimum Thickness of Vacuum Chamber

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  12. ORF

    Is the principle of minimum action applicable to nonholonomic systems?

    Hello Is the principle of minimum action applicable to nonholonomic systems? Why? If this question is already answered in this forum, just tell me, and I will delete this thread. Thank you for your time :) Greetings PS: My mother language is not English, so I'll be glad if you correct any...
  13. T

    Airplane pilot in a dive, net force, minimum speed, and normal force.

    Homework Statement A 65 kg airplane pilot pulls out of a dive by following at constant speed, the arc of a circle whose radius is 300 m. At the bottom of the circle her speed is 180 km/h. a.) Calculate net force on pilot at bottom or circle. b.) Calculate force exerted on pilot by seat at top...
  14. B

    Minimum Force of 86N to Move Two Blocks

    1. Homework Statement A 3.0-kg block sits on top of a 5.0-kg block which is on a horizontal surface. The 5.0-kg block is pulled to the right with a force ##\vec{F}##. The coefficient of static friction between all surfaces is 0.63 and the kinetic coefficient is 0.38. What is the minimum value...
  15. E

    Find Min Distance b/w Parabolas: 7/(4*root2)

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  16. BondKing

    Identifying local maximum, local minimum and saddle point.

    Indicate whether you think it is a local maximum, local minimum, saddle point, or none of these? My solution: Point P = Local Max Point Q = Local Min Point R = None Point S = Saddle I got a 75% for first attempt, so one answer is not correct and I am not sure which one isn't.
  17. W

    Is there a minimum speed in the universe besides the maximum speed of light?

    If there is maximum speed in the universe and we call it c, and it is not relative, is there also some ultimate rest state, an unrelativistic "0" speed? I sort-of know that there is none, because it would mean that there exists a non-relativistic frame of reference. But I can't quite grasp what...
  18. C

    Maximum and Minimum Values Inside Triangle

    Homework Statement Find the absolute maximum and minimum values of f(x,y) = y^2+x^2 -4x + 11 on the set D where D is the closed triangular region with vertices (8,0),(0,4) , and (0,-4) . Homework EquationsThe Attempt at a Solution [/B] fx = 2x - 4 fy = 2y Critical point = (2,0) The...
  19. C

    Maximum and Minimum Values (Trig)

    Homework Statement Find the 4 critical points f(x,y) = 5ycos(9x) closest to (0,0) Homework EquationsThe Attempt at a Solution fx = -45ysin(9x) fy = 5cos(9x) fxx = -45*9ycos(9x) fyy = 0 fxy = -45sin(9x) y=0 x=pi/18 (0,pi/18) (0,pi/18 + pi/2) (0,pi/18 - pi/2) (0,pi/18 + 3pi/2) Was not...
  20. R

    Minimum potential difference, Balmer series

    Homework Statement -A beam of electrons bombards a sample of hydrogen in its ground state. a) Through what minimum potental difference must the electrons be accelerated if only one line of the Balmer's series can be observed? Homework Equations V= h^2/2meλ^2 (V=potential difference)...
  21. zug

    What is the minimum wattage rating the resistor should have?

    Homework Statement [/B] The question "What is the minimum wattage rating the resistor should have?" comes after a previous problem I did to determine the maximum and minimum current when given a resistors tolerance. Here are all the values from that problem, some aren't really important but...
  22. M

    Minimum Distance between two curves

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  23. HaLAA

    How Does Speed Affect Car Stopping Distance?

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  24. D

    Minimum mean square error for two random variables

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  25. A

    MHB Question about minimum function and limit definition (proper)

    Hi, Suppose you want to prove $|x - a||x + a| < \epsilon$ You know $|x - a| < (2|a| + 1)$ You need to prove $|x + a| < \frac{\epsilon}{2|a| + 1}$ So that $|x - a||x + a| < \epsilon$ Why does Michael Spivak do this: He says you have to prove --> $|x + a| < min(1, \frac{\epsilon}{2|a| +...
  26. B

    Minimum Shear Stress in Hollow Circular Tube

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  27. W

    MHB Minimum Value of F(x): Does It Have a Maximum?

    Find the minimum value of F(x)=\int_{x^2-2x}^{0} 1/(1+t^2)\,d Does it has a maximum value? why?
  28. Satvik Pandey

    Block on wedge which is not fixed.Find minimum μ to keep the wedge at rest.

    Homework Statement Consider a wedge of mass m whose upper surface is a quarter circle and is smooth. The lower part is rough. A small block of same mass is kept at the topmost position and the initial position is shown in figure. The block slides down the wedge. Find the minimum value of...
  29. Fantini

    MHB Minimum time required to cover distance, beginning and ending at rest

    This is problem 1.16 from Kleppner & Kolenkow's Introduction to Mechanics. I didn't know whether to post it here or pre-university math. A sportscar, Fiasco I, can accelerate uniformly to $120$ mi/h in $30$ s. Its maximum braking rate cannot exceed $0.7g$. WHat is the minimum time required to...
  30. C

    Minimum volume of phase space

    In several books I have seen the statement that due to Heisenbergs principle no particle can be localized into a region of phase space smaller than ##(2 \pi \hbar)^3##. However, Heisenbergs uncertainty principle states that ##dx dp \geq \hbar/2## -- so a direct translation of this should imply...
  31. Albert1

    MHB Find Min Value: $a,b,c>0$ with $a+b+c=k$

    $a,b,c>0$ $a+b+c=k$ find:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
  32. michael879

    Deriving Møller's Relativistic Minimum Radius for Rotating Bodies

    Can someone either derive or point me to a derivation of Møller's formula for the relativistic minimum radius of a rotating body? I've been searching for about an hour and it's driving me crazy! The only "minimum radius" equation I've seen imposes the speed limit c on a classical rotating body...
  33. M

    First order minimum - Wave interference

    Hi :-) I'm currently learning about wave interference and came across two phrases I can't seem to find the meaning of; first order minimum and second order minimum. What do they mean an how are they used? I presume they are for the order of destructive interference or something?
  34. B

    Minimum kinetic energy of an Alpha particle confined to a nucleus

    Homework Statement http://puu.sh/bz0km/092f2d8781.png Homework Equations h^2/2ma^2 The Attempt at a Solution I've found the uncertainty of the kinetic energy through many means, and I'm confident that it's .052MeV. The easiest method that gave me this answer was: using...
  35. D

    Value of theta of first minimum

    Homework Statement A parallel beam of monochromatic light of wavelength 580nm is incidentally normal on a diffraction grating. The separation between successive lines on the grating is 0.7μm as shown in the figure. please refer to the photo , i am having problem with part (a) Homework...
  36. W

    Finding minimum variable's density function

    Homework Statement Let X and Y be independent and suppose that each has a Uniform (0,10) distribution. Let Z = min{X, Y }. Find the density fZ(z) for Z. Homework Equations The Attempt at a Solution i'm sorry but i really don't know how to handle this.
  37. V

    States with minimum energy for electrons in mag field with nonzero Lz

    Homework Statement For an electron in a uniform magnetic field, say B\hat{z} with no angular momentum, the Hamiltonian can be expressed as \hat{H}=\frac{1}{2m}\Big(\hat{p}_x^2+\frac{mω^2}{2}\hat{x}^2\Big)+\frac{1}{2m}\Big(\hat{p}_y^2+\frac{mω^2}{2}\hat{y}^2\Big) Which is equivalent to two...
  38. D

    Find Minimum Acceleration from Graph

    So this looks really easy, but I can't get the right answer. It asks you to find the minimum acceleration from the graph: http://s22.postimg.org/f5dutd5ap/pic.png. This is of course the point where the slope is the smallest, and this is from t = 3-4 s or t = 4-5 s, depending on whether you want...
  39. T

    How do you calculate the minimum film thickness?

    Hi, i am wondering how do you calculate the minimum film thickness between two plates, one smooth surface and another surface with a square microasperity. i am just very confused as to how do you determine the film thickness in between them. I have attached a paper which has a graph on...
  40. P

    Practical experience of minimum tension possible of a membrane

    I need to know a realistic minimum biaxial tension per unit length - in N/m - for the surface of a membrane which is 27.4 mm square - with surface density 0.0912 kg/m^2 - so that it could be vertical and remain tense so that mode shapes due incoming acoustic energy would remain intact (if the...
  41. J

    How wide is a single slit that gives its first minimum at 90°?

    Homework Statement How wide is a single slit that gives its first minimum at 90°? Homework Equations øp=p(λ/a) The Attempt at a Solution Not quite sure what the answer is looking for but I'd guess I would need to solve for "a." So... 90°=(λ/a) Is there any more...
  42. Dethrone

    MHB Finding the Minimum Value of a Summation with a Constant Term

    Given $a_1,...,a_n$, find the minimum value of $\sum_{i}^{n}(x-a_i)^2$ No idea how to do it. I was thinking maybe when $x-a_i=0$, but I think $x$ is constant so it won't work...unless the series $a_n$ is constant too...Tiny hint please :D?
  43. N

    Minimum requirements for international publications

    Hi, I'm just wondering what are the minimum requirements for an international publication/conference/proceedings. E.g. impact factor
  44. O

    The Theoretical Minimum book question

    I'm reading the book The Theoretical Minimum. I wonder if two one of the problems in the book can be thought of graphically. The mathematical solution didn't even occur to me, although it make perfect sense and probably should have been obvious. 1. Can you explain why the dot product of two...
  45. anemone

    MHB What is the Minimum of f(x) Without Using Calculus?

    Without using the calculus, determine the minimum of $f(x)=(x^2-6)(x^2-8)(x^2-10)(x^2-12)+(6\cdot 8\cdot 10\cdot 12)$.
  46. K

    Are There Really Slow Moving Electrons?

    What is the minimum speed at which electrons can be emitted? Why is it I never heard of really slow moving electrons, like 10 meters per second slow?
  47. N

    Minimum solenoid activation time?

    Heey guys, I have a problem. I want to calculate the minimum time i have to activate a (latching)solenoid so that its final velocity is 0 m/s when it hits the inside. Well I know the following things: F= m*a I know the force and the mass, so a can be calculated. V = a*t. the average...
  48. gfd43tg

    Equivalent minimum linear program

    Homework Statement Suppose you are given the following affine minimization problem. \begin{align*} \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in}&c^Tx+3c^Tc \\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*} Which of the following is an equivalent linear minimization problem...
  49. B

    MHB How do I decide this total cost and how do I make a minimum estimate?

    Principle: 10K Interest: 10% compounded annually Time: Over five years What is the total cost of all of this factored in? Could I get the formula so I can use it to apply to a real problem?
  50. gfd43tg

    Minimum value of the expression

    Homework Statement Problem is posted as imageHomework Equations The Attempt at a Solution Hello, I am having some confusion over what is meant by 'type in the boxes the minimum value of the expression'. Does that mean take the derivative of the function? Or does that mean the value at which...
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