# What is Maximum: 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. ### Show proof of point C in the given problem that involves Polar equation

c Parts (a) and (b) are okay ... though the challenge was on part (a) My graph had a plot of r on the y-axis vs θ on the x-axis). The sketch of my graph looks like is shown below; I suspect the ms had θ on the x-axis vs r on the y-axis. I used the equation ##r=\sqrt{\dfrac {1}{θ^2+1}}##...
2. ### Exploring the Non-Existence of a Maximum in the Set (0,2)

For this, I am trying to understand why the set ##(0,2)## has no maximum. Is it because if we say for example claim that ##a_0 = 1.9999999999## is the max of the set, then we could come along and say that ##a_0 = 1.9999999999999999999999999999## is the max, can we continue doing that a...

4. ### Calculate Max Resistance for 2 Parallel Wires

Hi, I am not sure if I have calculated the task here correctly: Based on the drawing, I now assumed that the two resistors are connected in parallel. The total resistance can then be calculated as follows ##\frac{1}{R_T}=\frac{1}{R_1}+\frac{1}{R_2}##. Since the two wires are made of the same...
5. ### Cubical Pressure Vessel - Maximum Pressure Computation

Hello everyone - this is my first post on the forum and this would be for a real world application. I have no mechanical engineering experience. We plan to build a cubical aluminium (6082 grade) pressure vessel. The total external dimensions are 540mm x 400mm x 75mm. The aluminium wall...
6. ### A Minimum-variance bound for the extended maximum likelihood estimation

I am fitting a mass spectrum using pdf(M)=Ns×S(M)+Nb×B(M; a, b) to determine the yield with the extended maximum likelihood fit, where Ns and Nb are the number of signal and background events, S(M) is the function for the signal, B(M;a, b) is the function for the background with parameters a and...
7. ### How do manufacturers determine the 'rated' quantities for motors?

I'm confused with the term "rated". I checked this webpage though I am not confident about it's reliability. Context: Motor ratings etc. Does it mean maximum? Maximum in what regard? Heat produced/temperature? Is there a more reliable source on how manufacturers determine the 'rated'...

9. ### I Heating a tungsten filament to find out the maximum brightness emitted

I was taking notes from a lecture on Quantum Physics and during the introduction, they gave an example of what led to the discovery of Quantum Physics: The electric bulb example where the brightness and colour of light depended on the temperature of the filament(see...
10. ### A Does the Maximum Lyapunov exponent depend on the eigenvalues?

I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
11. ### A Landau's Maximum Mass Limit Derivation in Shapiro & Teukolsky (1983)

In Section 3.4 of Shapiro & Teukolsky (1983), a simple derivation, due to Landau, of the maximum mass limit for white dwarfs and neutron stars is given. I will briefly describe it here and then pose my question. The basic method is to derive an expression for the total energy (excluding rest...
12. ### Engineering Maximum modulation factor of a two tone signal

I'm stuck on this because v is a 2 tone signal, so it's not as simple as Am/Ac. The teacher said I will need to differentiate it and equate it to zero, which I thought made sense. Differentiating v gives me: [- mwsin(wt) - mwsin(2w*t)], so there's still unknown variables. I don't know how I'm...
13. ### What Is The Maximum Force The Locust Exerted on The Plate?

At the beginning, I just looked for the highest point in the graph, which is approximately 90 \frac m s^2. Then I plugged it in the formula F = ma and got the force equaled to 0.045 N. However, when I looked back, the graph is about the change in acceleration. So really, I'm dealing with...
14. ### Maximum angle reached by a cube placed inside a spinning cylinder

I am trying to solve a problem where we have to find the maximum angle before a cube starts sliding when said cube is placed inside a spinning hollow cylinder (the cylinder is placed horizontally). The radius of the cylinder is 0.4 m, the coefficient of static friction between the cube and the...
15. ### Loader Boom maximum bending moment

TL;DR Summary: Finding maximum bending moment on a boom I am trying to solve the problem stated in the picture below. The answers are given. I've manage to find the Normal and Shear Forces but I'm stuck at the bending moment. According to my analysis the maximum bending moment is +7208,1...
16. ### Evaluating minimum and maximum values with calculations

Hi! I know this may seem like a silly question but I really just want to make sure I understand this correctly. I've already calculated the minimum and maximum values for time and height: t min = 0.58 s t max = 0.68 s y min = 1.98 m y max = 2.02 m To calculate the minimum average speed, would...
17. ### POTW Expected Value of a Maximum

If X and Y are independent standard Gaussian variables, find the expected value of the maximum of X and Y. Note: This problem is due to @Office_Shredder.
18. ### Calculate the best angle for maximum light dispersion through a medium

I first thought that the angle would have to be maximum when it is closest to the critical angle for total internal reflection. From my lectures the equation for the critical angle is ##\theta _1>\ sin ^{-1} \left( \frac {n_2} {n_1} \right),## so as ##n_2 = 1##, we have ##\theta _1=\sin...
19. ### Distance between the first and third maximum intensity detected

My answer is 0.5 m (same as wavelength) but answer key is 1 m. I thought the maximum intensity occurs at antinode and distance between adjacent antinodes is half of wavelength so the distance between first and third antinode is one wavelength. Where is my mistake? Thanks
20. ### For a Pendulum: Knowing Acceleration Find Maximum Angle

PS: By the way today I had exams in Physics and this problem was the first one I had to solve :p (unlucky) The question was to find the maximum angle θ that the pendulum can reach if we know that the magnitude of the acceleration is the same when the mass is located in the highest and the lowest...
21. M

### I Maximum likelihood to fit a parameter of this model

Hi PF! Given random time series data ##y_i##, we assume the data follows a EWMA (exponential weighted moving average) model: ##\sigma_t^2 = \lambda\sigma_{t-1}^2 + (1-\lambda)y_{t-1}^2## for ##t > 250##, where ##\sigma_t## is the standard deviation, and ##\sigma_{M=250}^2 =...
22. ### Maximum duration of Solar eclipses

The result I get is 2 hours and 10 minutes. My reasoning is down here. But! Checking the map for the 2024 Solar eclipse, https://www.timeanddate.com/eclipse/map/2024-april-8, in many locations you can see a partial eclipse of over 2 hours 40 minutes. What is the main source of error here? Here...
23. ### Engineering Calculating maximum flux density

Hi everyone, I have a EE problem that I need to sort out for alternating voltage. I have to find out the maximum flux density. B_max= integral from 150 degrees to 30 degrees (u/(2NA) dt is my problem. I have a hard time to integrate this since I am to integrate with time and not degrees or...
24. ### Engineering What is the impact of ADC quanta on op-amp offset error?

Hello, i have questions from a book that I'm reading, it's full of interesting problems about Electrical Engineering. For now i want to ask about op-amp offset error. I will post a copy of the problem here. We can talk about it, i have an idea about offset error so i can share it here later...
25. ### Find surface of maximum flux given the vector field's potential

The vectorfield is $$A = grad \Phi$$ $$A = x^2 + y^2 + z^2 - (x^4 + y^4 + z^4 + 2x^2y^2 + 2x^2z^2 + 2y^2z^2)$$ The surface with maximum flux is the same as the volume of maximum divergence, thus: $$div A = 6 - 20(x^2 + y^2 + z^2)$$ This would suggest at the point 0,0,0 the flux is at maximum...
26. ### Planar Motion — Maximum acceleration of a crate up an incline before tipping over backwards

Correct answer is (b) 0,620 ms2 My attempt: FBD Equation 1. ∑Fx=0=−ma−Wsin⁡(15)+f N=Wcos⁡15 f=Nμs=379,03N Equation 1. ma=f−Wsin⁡15 15a=379,03−203,1 a=2,20ms2 Equation 2. ∑MG=0=−Nx+f1,5 x=0,75m
27. ### How can I calculate the maximum impact force on the helmet?

I am studying the force that helmets have to withstand. If I drop a 5kg head model and helmet at 1.064m on an anvil and the impact time is 0.005 seconds, how do I get the 'maximum' impact load on the helmet?
28. ### How to find the positive maximum value of a function

This is the code that i wrote Clear["Global`*"] Z = 500; W = 100000; G = 250; H = 100; K = 0.5; T = 30; L = 4000; P = 5; S = 2.5; Y = 1; A = 0.1; V = 2.5; J = 8000; f[x_] := 1/ x {(J*Z*x*(2*Y - x))/( 2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/ 3 + (H + T*S +...
29. ### Maximum charge on the plates of a capacitor

What I have done: The electromotive force due to Faraday's Law is: ##\mathcal{E}=-\frac{d\phi(\vec{B})}{dt}=\frac{d}{dt}(Ba^2)=a^2\frac{dB}{dt}=-10^{-4}V.## In the circuit, going around the loop in a clockwise fashion: ##\oint_{\Gamma}\vec{E}\cdot d\vec{l}=-\frac{d\phi(\vec{B})}{dt}\Rightarrow...
30. ### Maximum bending moment and maximum deflection of the spring?

How can I find the maximum bending moment and maximum deflection for a spring? It would be very helpful if you could explain the specific procedure and formula in an easy-to-understand manner. that's all, thank you very much.
31. ### High-Power LEDs -- Maximum 'Lumen-Density'?

Light-emitting diodes (LEDs) are replacing traditional lighting technology -- such as halogen -- in high-power lamps, as LEDs are much more efficient (Lumens/W) and longer-lasting. A commercial example of high-power LED flood lights: High Power 3000W LED Flood Light - Lsleds NOTE: The above...
32. ### Find Projectile Flight Time Given Only Maximum Height

I've been trying some online projectile problems. Specifically, I was using this one on master difficulty, looking at row e. It uses a random number generator; I shared the data I received in the Homework Statement above. According to the help section, you can solve this with the formula...
33. ### Value of p for dp/dt to be maximum

To find maximum of ##\frac{dp}{dt} \rightarrow \frac{d^{2} p}{dt^{2}}=0## $$\frac{k}{c}p'(t)(1-p(t)^{c})-\frac{k}{c}p(t)(c.p(t)^{c-1}).p'(t)=0$$ $$p'(t)=0~\text{or}~p(t)=\left(\frac{1}{c+1}\right)^{\frac{1}{c}}$$ For p'(t) = 0 → p(t) = 0 or p(t) = 1 So I have 3 values of p(t), how to know...
34. ### Maximum Deflection Angle In Elastic Collision

I started by writing the equation v1i + v1f = 2v and then drawing a triangle with v1i, v1f, and 2v as the three sides. Then I used the Law of Cosines to solve for cos theta but this did not lead to a solution. Could I have a hint on how to begin? Thank you!
35. ### Finding the maximum size of an Initial Value Problem coefficient

The following IVP diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39 Describes the tempetatur T in celcius at the time x of a tub filled with water. A tub which is filled with hot water at rate of v l/min. Lets say I am told that a guy takes a 40 min bath, and during those 40...
36. ### Estimating maximum percentage error using partial differentiation

Attempt at question No. 1: ΔD = ∂D/∂h * Δh + ∂D/∂v * Δv ∂D/∂h = 3Eh^2/(12(1-v^2)) ∂D/∂v = 2Eh^3/(12(1-v^2)^2) Δh = +- 0,002 Δv = 0,02 h = 0,1 v = 0,3 ΔD = 3Eh^2/(12(1-v^2)) * Δh + 2Eh^3/(12(1-v^2)^2) * Δv Because the problem asked for maximum percentage error then I decided to use the...
37. ### Calculating Maximum Height: Confusion & Solutions

I am calculating it like this: 𝑦=ℎ0−0.5𝑔𝑡^2=0→ℎ0=0.5𝑔𝑡^2→𝑡=sqrt(2*ℎ0/g) 𝑥=𝑣0*𝑡→ substituting t →𝑥=𝑣0*sqrt(2*ℎ0/g) 𝑑𝑥/𝑑ℎ0=𝑣0/(𝑔*sqrt(2*h0/g))=0 for maximum ℎ0=0. confused. can someone tell me how I am calculating this wrong?
38. ### Why is my calculation for a rocket's maximum range incorrect?

the first question was At what launch angle will the rocket achieve its maximum range? and i found the right answer it was 42.186deg. but now i don't understand what I am doing wrong i put v0,a,g in the equation of range and its wrong and i don't undertand why here is what i try...
39. ### Safety of plastic spoons, etc. below their advised maximum temperature?

I like using polypropylene dishware e.g. spoon, and storage box. I can see that from its label sometimes the maximum temperature recommended is 110 degree C or 120 degree C but it usually doesn't go away from this range. By common sense, I will never heat the polypropylene dishware directly but...
40. ### A Max Force Debate: 2 vs. 3 Papers in Favor/Against

The number of papers on maximum force c^4/4G and similar limits in general relativity is growing. Last year there were A. Jowsey et al, arXiv:2102.01831 V. Faraoni, Phys Rev D 103, 124010 (2021) C. Schiller, Phys Rev D 104, 068501 (2021) V. Faraoni, Phys Rev D 104, 068502 (2021) C. Schiller...
41. ### Car's maximum acceleration on a road is proportional to what?

Since the car starts from rest it’s accelerating. So, $$F_a-f_k=ma$$ $$F_a-\mu mg=ma$$ $$\frac{F_a-\mu mg}m=a$$ Now from second eqn, ##s=ut+\frac12at^2## $$s=\frac12\frac{F_a-\mu mg}mt^2$$ $$\frac{2sm}{F_a-\mu mg}=t^2$$ $$\sqrt{\frac{2sm}{F_a-\mu mg}}=t$$ I don’t think I am getting any where!
42. ### I Unbalanced forces acting on a ball at maximum altitude

When a ball projected vertically upright reaches its maximum height is it instantaneously stationery (calculus provides for instantaneous events) before it starts its downwards journey? is gravity acting on the object at that instant? Newton provides that an object at rest must be subject to...
43. ### B Maximum Force on transom of a boat under acceleration

I'm trying to calculate the maximum Force in PSI applied to the transom of a planing boat under acceleration using F=ma. boat mass is 300kg accelerates from 0 to 10m/s in 6 seconds F=ma -> 500N = 300kg *(10m/s/6s) = 112.4045lbf the contact area between the boat transom and motor is 1000 mm2 =...
44. ### Maximum charge on a spherical capacitor

The electric field is the one generated by the charge ##+Q## on the inner sphere of the capacitor, which generates a radial electric field ##\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}## which, due to the presence of the dielectric, become...
45. ### Analyzing a Combination of Beer+Lower Arm for Maximum Acceleration

Hello guys, to analyse the above-described situation I have opted for considering the body "beer+lower arm" as a whole, therefore computing the new position of the COM for the correct FBD and equations of motion. With some uncertainty on the statement of the problem, I have interpreted the...
46. ### Maximum detectable distance between magnet and magnetometer

Hi Dear, I have a single-component (Z-Axis) magnetometer with 60nT resolution. I want to calculate the maximum distance that this magnetometer can detect an N52 magnet with a volume of 1 cm³. So I used the following formula and inserted the values, and finally calculated the Z...
47. ### B Exploring the Possibilities of a Higher Cosmic Speed Limit than c

I think when the speed of light was measured (and predicted from Maxwell's equations) that the assumption was made that this speed was a cosmic speed limit Suppose that the cosmic speed limit was higher than c (not infinite) and that perhaps another form of radiation traveled at that speed...
48. ### Find the maximum value of the product of two real numbers

Using the inequality of arithmetic and geometric means, $$\frac {x+y}{2}≥\sqrt{xy}$$ $$6^2≥xy$$ $$36≥xy$$ I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.
49. ### Calculate the speed of car for maximum economy

This is the problem, This is the textbook solution ; i think the textbook solution is not correct! My thought, ##S_{max}## will be given by $$\frac {dy}{dS}=0$$ $$\frac {dy}{dS}=\frac {-1}{400}S+\frac {1}{5}=0$$ $$S=80$$ It is maximum economy because ##y{''}<0##, i.e by use of the second...
50. ### B Grain direction for maximum tear resistance for a wire bound book

Paper has a grain direction – the direction of the long fibres. For several reasons, publishers always print hardcover books with the grain of the paper in the same direction as the fold. I'm working on a book that will be 2:1 wire bound (two holes to the inch). I was wondering what the grain...