Maxima Definition and 155 Threads

Ok i have, ##f_x= 3x^2-y-1+y^3## ##f_y = -x+3xy^2-4y^3## ##f_{xx} = 6x## ##f_{yy} = 6xy - 12y^2## ##f_{xy} = -1+3y^2## looks like one needs software to solve this? I can see the solutions from wolframalpha: local maxima to two decimal places as; ##(x,y) = (-0.67, 0.43)## ...but i am...

For this problem, Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)? Many thanks!

Hi, Hi, Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$ But when I read this graph, I couldn't guess that the stated function have a...

Question: Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone. Answer: Let r and h be the radius and height of the right circular cylinder inscribed in a given cone of radius R and height H. Let...

Consider the following theorem: Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.## The intuition is that local maxima and global maxima coincide for...

Good day, I have a question regrading how to find the absolute minima maxima of a function , I understand that first we need to calculate the Hessian Matrix to find the relative minima /maxim but after we need to check the borders of the region ( a rectangle in our case) for example we put x=-2...

When the crystal above is illuminated with light of intensity ##I_0##, what is the intensity at the central maximum? (The picture shown above is a 4 x 4 unit cell of the crystal) The answer is ##(\frac {5} {16} )^2 I_0##. Why? Apparently, Electric field is proportional to the transparent area...

I drew a diagram in the attached files as well, but the the scenario seems to be the same as the double slit experiment, but I don't understand why the answer contains cos instead of sin.

I am switching from Mathematica to Maxima. Although both of them are symbolic computation programs, I think it is better to start studying Maxima's language from the basic stuff. Could someone recommend some good video tutorials to me? Thanks.

The ranked order from most to least effectively reducing the widths of the primary maxima is: A>D=C>B. I know that doubling the wavelength of incident light will double the width (y) of the fringe. This means that doubling the frequency of incident light will reduce the fringe width (y) by half...

I honestly do not have any idea regarding this sum. Any help will be appreciated:)

Hi PF! I have data that I need to interpolate (don't want to go into details, but I HAVE to interpolate it). I'm trying to find the local maximas on a given domain. I've looked everywhere and still haven't been able to do it? Seems most people work with NDSolve, but I don't use that function...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need yet further help in fully understanding the proof of Proposition 8.14 ...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some further help in fully understanding the proof of Proposition 8.14 ...

Homework Statement: In a double slit experiment let d=5.00 D=30.0λ. Estimate the ratio of the intensity of the third order maximum with that of the zero-order maximum. Homework Equations: interference diffraction i guess the goal is this equation ##I_{(\theta)}=I_0 \times(cos^2\beta)\times...

Does anyone know how to get Maxima to solve the Kerr metric? I enter the terms for that metric that I found on Wikipedia. It tries to print out the Einstein tensor (covariant, leinstein(true)) and the expressions are so long that it literally locks up my computer. And isn’t the Kerr metric a...

Homework Statement given the formula m=n*e^(-nt) show that the maximum of this curve is at m=1/(t*e^(1)). 2. The attempt at a solution I can show this graphically but I am curious if it is possible to do it by hand?

Homework Statement Find out the quotient derivative i.e. the derivative of polynomial upon polynomial and then find the minima and maxima.[/B] ##W\left(z\right)=\frac{{4z+9}}{{2-z}}## Homework Equations ##\left( \frac{f}{g} \right)' = \frac{f'\,g - f\,g'}{g^2}## The Attempt at a Solution...

Greg Bernhardt submitted a new PF Insights post Solving Einstein's Field Equations in Maxima Continue reading the Original PF Insights Post.

When observing interference patterns, one will notice that the maxima from a double slit are more intense and farther apart than the maxima of a single slit. Likewise, the maxima from a diffraction grating are more intense and farther apart than maxima of a double slit. Why is this? I...

Homework Statement f(x,y)=2x^3+xy^2+5x^2+y^2 Homework EquationsThe Attempt at a Solution fx(x,y)=6x^2+y^2+25x fy(x,y)=2xy+2y fxx(x,y)=12x+25 fyy(x,y)=2y+2 fxy(x,y)=2y I found the partial derivatives from the equation. I am stuck at finding the critical points from the first two fx(x,y) and...

I'm trying to use Maxima to examine the error in a Fourier series as the number of terms increases. I've figured out how to produce a Fourier series and plot partial sums, but this has me stumped. If anyone experienced with the Maxima CAS has some insight into this, I would greatly appreciate...

Homework Statement A diffraction grating is made up of slits of width 300 nm with separation 900 nm. The grating is illuminated by monochromatic plane waves of wavelength l 600 nm at normal incidence. How many maxima are there in the full diffraction pattern? Homework Equations mλ = dsinθ The...

Let's say we are using a spectrometer to observe the order spectrum of mercury, a polychromatic source. Each order contains a an array of colour bands. In the case of the following image: http://imgur.com/a/bnm9r , I was wondering if each Fraunhofer envelope represented one order, (m = 1 , 2 ...

I need to find the max/min of a function on an interval. The function is f(x)=x+cos(x) and the interval is <-PI,2pi> There is an attached solution but I do not understand how to arrive at the given solution (see screenshot). I would personally just take the derivative as F'(x)=1-sin(x) However...

I’ve been wachting a video here on interference maxima when light passes a grating and 3 questions came up. 1. The lecturer shows the following picture: He mentions that each purple line from each slit (distance ##d## apart) that meet at the same point on the right side (for constructive...

Hi there I'm prepping for a big test tomorrow and I'm really struggling with this question:If f′′(x)≥−1, x belongs to (−15,15), and f′(1)=3, find the interval over which x is definitely increasing.I'm struggling with substitution because I just don't seem to have enough values. Is there a...

I have recently been doing some work that involves long, messy manipulations with lots of geometric series. A typical such series, which would only be one of a number of such terms in a formula, is: $$ \sum_{t=h+1}^{T-h} \left(1-(1-\theta)^{T-t-h+1}\right) \\ $$ It's not difficult to simplify...

Hey! :o We have the function $f(x_1, x_2)=2-x_1-x_2$ and we want to check if it has maxima or minima under the constraint $x_1^2+x_2^2=8$. Since we cannot solve for one variable at the equationof the constraint, we have to use the Langrange function, right? (Wondering) I have done the...

Homework Statement Find at which angles θ the interference picture that appears on a distant screen made by three thin slits separated by distance d and enlightened by a source of wavelength λ (see figure) a) Shows its maxima. b) Shows its minima. multiple slit diffraction, d, ##\lambda##...

In general, materials are either direct or indirect materials. For direct materials, both the valence band maxima and the conduction band minima are at k = 0. For indirect materials, only the valence band maxima is at k = 0 and the conduction band minima is at some other non-zero value of k. In...

Homework Statement Do the maxima created by a diffraction grating all have the same intensity? Explain. Homework Equations mλ = wsinθ (?) The Attempt at a Solution According to the back of my textbook (Nelson Physics 12), the answer is no. I believe the answer may have to do with the fact...

Homework Statement So I want to prove that the expression 20Cr×0.1r 0.9(20-r) reaches maximum value for r=(0.1)×20=2 Homework EquationsThe Attempt at a Solution I can prove it by trial and error but can't differentiate the expression because nCr isn't continuous.

Homework Statement Light of wavelength λ = 535 nm shines through two narrow slits which are 670 μm apart. What is the maximum number of interference maxima which could conceivably be observed (assuming that diffraction minima do not extinguish them and the screen is arbitrarily large)?Your...

Homework Statement Find the local max, min, and saddle point for the function: f(x,y) = 2x^2+3xy+4y^2-5x+2yHomework EquationsThe Attempt at a Solution I've taken the two partial derivatives Fx = 4x + 3y - 5 Fy = 3x + 8y + 2 I know that the critical points will sit where both of theses partial...

Homework Statement Find where θ is the biggest (largest) I'll have the picture of the problem included below (pic:1) Homework Equations (x-q)2+(y+5/2)2=r2 answer x= 2 The Attempt at a Solution Hi, so my prefesor gave me this problem and told me to try to solve it. We already did this problem...

Hello Forum, I think I am clear on how to find the maxima and minima of function of one or two independent variables like f(x) or f(x,y). What if the function had 5 independent variables, i.e. f(x,y,z,a,b)? What is the best method, even numerically? Should we find the partial derivatives of f...

Write the following formulas: a) The minimum perimeter of any triangle (abc) only known heights corresponding to the sides a and b. b) The maximum height and minimum corresponding to the side b of any triangle (abc) only known the value of its perimeter and height corresponding to the side a.

Third question about wave optics in two days, hope it's not too much. When we look at the intensity of light on the screen in a double-slit interference experiment (assuming negligible diffraction), we find something like this : When we look at the intensity of light on screen in a...

Homework Statement A grating has a line density of 1010 cm−1, and a screen perpendicular to the ray that makes the central peak of the diffraction pattern is 2.5 m from the grating. If light of two wavelengths, 590 nm and 680 nm, passes through the grating, what is the separation on the (flat)...

I was working on derivation to find the witdh of principle maxima in n slit diffraction Angular width=2Δθm=2λ/(Ndcosθm) Where d:distance between slits θm:angle at which mth order principle maxima is present. N:no. Of slits #in the diagram given below why the minima just adjacent to mth...

Homework Statement Please if anyone could help me with this question . Two wires AC and BC are attached to the top of pole CD . The force exerted by the pole at C is vertical and the 2.5 kN force applied at C is horizontal and makes an angle α with the direction of the x-axis . Find the...

I have the following function: f(x,y) = xye(-x2-y2) I am trying to find all the maximum and minimum points. I have started off by finding the partial derivatives of the function and equation them to zero. df/dx = ye(-x2-y2)(1-2x2) = 0 df/dy = xe(-x2-y2)(1-2y2) = 0 However, i am stuck for...

I'm working on a homework assignment and I'm completely stuck on this last problem. I'm not even sure what ∇KE=0 even means to begin with. I understand the difference between indirect and direct bandgap but I'm just confused on how to find the maxima and minima to determine this. Any help would...

Homework Statement A single slit diffracts laser light of wavelength 650 nm onto a screen 2.00 m away. The distance between the two first-order maxima on either side of the central peak is 2.05 mm. How wide is the slit? Homework Equations a=width of slit y=distance from central max to specific...

Homework Statement Given critical points (3,-4) and (6,0); interval of increase (3, infinity); interval of decrease (-infinity, 3), find the local maxima/minima and sketch the graph. Homework Equations No relevant equations are given, I believe it's a simple sketch the graph. The Attempt at...

for: solve([k^2=(xa-xd)^2 + (ya-yd)^2,k^2=(xb-xd)^2 + (yb-yd)^2,k^2=(xc-xd)^2 + (yc-yd)^2],[xd,yd]); the output is simply : [] I want it to solve the system : k2=(xA-xD)2 + (yA-yD)2 k2=(xB-xD)2 + (yB-yD)2 k2=(xC-xD)2 + (yC-yD)2 for xD and yD. I already solved it by hand but wanted to test...

Homework Statement A uniform rod of mass M and length L can rotate around point P which is at position x from one end of the rod as shown in the figure. The rod is gently placed on a rough horizontal surface that has a friction coefficient μ. and at t=0 starts rotating with angular velocity...

I am clueless. Here's the model I tried to follow: g(a):=a^2-1; f(a,b):=a+g(a)-b; f(a,b); With output −b+a^2+a−1 Which is 1)Define a function in terms of some variable(s). 2)Define another function in terms of some variable(s) and in terms of the function defined in 1). 3)Evaluate the function...