Critical points Definition and 144 Threads

Homework Statement I have the scalar function T(r)= 2x^{2} - 4x +y^2 +4y-3z^{2} (r is obviously a vector) I have the critical point (1,-2,0) but I'm not sure how to work out if its a min/max/saddle. I'm familiar with doing it for functions in 2 variables.

Homework Statement See figure. Homework Equations N/A. The Attempt at a Solution Part A: The volume is, xyz = xy(1 - x^{2} - y^{2}) Critical points: f_{x} = y-3x^{2}y-y^{3} = 0 f_{y} = x -x^{3} - 3xy^{2} = 0 Part B: This is where I get confused, how do I...

Homework Statement Find all critical points and classify them Homework Equations f(x,y) = sin(x)sin(y)sin(x+y) 0<=x,y<=Pi The Attempt at a Solution fx=sinysin(2x+y) and fy=sinxsin(2y+x) Therefore critical points are at: x=Pi/3(2n-m) , y=Pi/3(2m-n) where n>=1, m<=2...

Hi. I have critical points are plotted on contours, I just need to construct Reeb graph for it. Any body has got idea to do it? I need algorithms or at least ready software. Please help me. Thanks. Lutas

Homework Statement Had an optimization test today and had a problem with this question. I needed to find the stationary points of this function: f(x)=\sqrt{x^{4}+1}-x+3 The Attempt at a Solution So you differentiate that puppy with some chain rule for the square root part and tidy up you...

Homework Statement Find the critical points of this function and determine wether they are local maxima, minima or saddle points... f=\frac{1}{x} + \frac{1}{y} + xy The Attempt at a Solution start off by partially differentiating and setting to zero for x and y: \frac{\partial...

Hello, I have solved for the critical points using the gradient, and I have solved for the Hession, which yields a 2x2 matrix. I have plugged in my critical points into the gradient. Now, do I apply the same rules as in linear algebra where I find the determinant and trace to calculate...

Homework Statement Let f(x,y)=Ax2+E where A and E are constants. What are the critical points of f(x,y)? Determine whether the critical points are local maxima, local minema, or saddle points. 2. The attempt at a solution First I found the first partial derivatives with respect to...

i have the following problem: find the critical points of: P = (x_{1} - 1)^{2} + (x_{n})^{2} + \sum(x_{k+1} - x_{k}) the bounds of the sum are from i = 1 to n-1. so i differentiate P with respect to x and i set it equal to zero, and i eventually get the expression: \sum(x_{k+1} -...

Hi, i was wondering if someone could please help to find and classify the critical points of : f(x,y) = (x-y)^2 What i know: I got fx = 2(x-y) and fy = -2(x-y) and in order to find the critical points we need to solve: 2(x-y) =0 -2(x-y) = 0 so if x =y then the above hold. where...

I was given f(x,y,z) = x4 + y4 + z4 - x2 - y2 - z2. I found that (or least I think it's these) x = 0 & \pm1/\sqrt{2}, y = 0 & \pm1/\sqrt{2}, z = 0 & \pm1/\sqrt{2}. What I'm stuck with is exactly how much critical points are there, by the looks of things there are a few but I'm not too sure...

Homework Statement Find the critical points of the function f(x, y) = sinx + siny + cos(x+y) where 0<=x<=pi/4 and 0<=y<=pi/4 Homework Equations First and second order partial derivative of f(x, y) The Attempt at a Solution To find the critical points, I first find the first...

Homework Statement Determine function f(x)=\frac{\sqrt[3]{x-4}}{x-1} critical points and find max and min value in given interval [2; 12] The Attempt at a Solution 1) I've to find derivative: f(x)'=\frac{(\sqrt[3]{x-4})' (x-1)-(\sqrt[3]{x-4})(x-1)'}{(x-1)^2}=...

Homework Statement Find all critical points of f(x,y) = 12xy-x^2 y-2xy^2 and test them for relative extrema using the 2nd partials test. Homework Equations The Attempt at a Solution {0=fx(x,y)=12-2x-2 10x-2 x=5 {0=fy(x,y)=12-4y y=3 Critical point: (5,3)...

1. Homework Statement [/b] What is/are the critical points of Kirchoff's Law: L\left(\frac{di}{dt}\right) + Ri = E The Attempt at a Solution I solved the differential equation above and got the following solution (which I verified to be correct): i = \left(\frac{E}{R}\right) +...

Homework Statement Find and classify all critical points of f(x,y) = x^3+2y^3-3x^2-3y^2-12y Homework Equations The Attempt at a Solution So I've gotten to Fx = 3x^2 - 6x = 0 Fy = 6y^2 - 6y - 12 = 0 x=0, x=2, y=-1, y=2 Now I can't remember how to find the critical points...

Homework Statement f(x,y)=xy+(144/x)+(12/y) f has a relative minimum at ( , , ) Homework Equations Partial derivatives The Attempt at a Solution fx=y-(144/x^2) fy=x-(12/y^2) I setted these to zero. Multiplies both sides so there is no fraction. Came out to be...

I need help finding the critical points of this equation: f(x) = xln(x) I found f'(x) to be ln(x)+x, but I don't know how to solve for x when setting f'(x) to 0 to find the critical points. I know I can find the zeros of f'(x) graphically on a calculator, but I need to know how to do it...

Homework Statement Find the critical points of f(x,y)=x+y^2 subject to the constraint g(x)=x^2+y^2=1 Homework Equations \nabla f=\lambda\nabla g g(x,y)=1 The Attempt at a Solution f_x=1=2\lambda*x\Rightarrow x=\frac{1}{2\lambda} f_y=2y=2\lambda*y\Rightarrow...

Homework Statement Find local max/min, and saddle points (if any) of f(x,y)=x^2+y^2+x^2y+4 This should be simple, but I am having algebra-block on solving the partial derivatives to find the critical points. f_x=2x+2xy=0 (1) f_y=2y+x^2=0 (2) If I multiply the second equations by...

x' = y(1 + x - y2) y' = x(1 + y - x2). C.P. => (x,y) s.t. f(x) = 0 f(x) = x' = 0 => y = 0 f(x) = y' = 0 => x = 0 => (0,0) is c.p. plug x = 0 into x': x' = y(1 - y^{2}) => y = 0, y =\ ^{+}_{-}1 => (0,1), (0,-1) are critical points. plug y = 0 into y': y' = x(1 -...

Sketch the following function, showing all work needed to sketch each curve. y = \frac{1}{3 + x^2} The question is asking for all the work done to find x and y intercepts, vertical, horizontal and slant asymptotes; critical points and points of inflection, i have completed the question...

Homework Statement y1'= −4*y1+y2+y1*y2 y2'= −2*y1−y2+y1*y1 Determine the three critical points of the system and their type of local phase portrait (stable node, unstable, saddle point, spiral, center, no node) Hence I need to get three critical points (x1,y1), (x2,y2) & (x3,y3) and their...

I have to find the critical points for the partials: f_x = y/3(24 - 12x - 4y) = 0 f_y = x/3(24 - 6x - 8y) = 0 I get y = 0 and x = (6-y)/3 for x in the first partial. How am I supposed to proceed? If I plug these into the secon, it gets nasty. Can anyone demonstrate how this is done...

I have two problems. I posted the first problem before but I still can´t solve it. Homework Statement Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3 Homework Equations - The Attempt at a Solution df/dx = y + z +3x^2, df/dy = x + z + 3y^2...

Homework Statement Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3 Homework Equations - The Attempt at a Solution df/dx = y + z +3x^2, df/dy = x + z + 3y^2 and df/dz = x + y + 3z^2 a point x is a critacal point if the gradient equals 0...

can anybody tell me what the critical point of xe^x is? When I try putting it into my calculator, it just shows a line staring at zero with an asymptote at x=1.

Homework Statement Find the singular points for: x' = ax - bxy y' = bxy - cy Homework Equations The Attempt at a Solution ax - bxy = 0 bxy - cy = 0 implies x = 0 or y = a/b y = 0 or x = c/b the the critical points are (o,o) or (c/b, a/b) why is (0, a/b) and...

I was wondering if we include asymptotes as criticals: For example in y = x sqrt(4-x^2) when u find the critical points it gives u root 2 and negative root 2 But when I draw the graph I am missing the 2 and -2 which were the asymptotes which help define the graph? So are they needed...

Homework Statement Consider the plane dynamic system \dot{x} = P(x,y), \dot{y} = Q(x,y) with the condition that O(0,0) is a critical point. Suppose P(-x,y) = -P(x,y) and Q(-x,y) = Q(x,y). Is the critical point (0,0) a center? Why? The Attempt at a Solution I know that for (0,0) to be a...

Homework Statement Find all the critical points: \frac{dx}{dt} = 1 - xy \frac{dy}{dt} = x - y^3 Homework Equations \frac{dx}{dt} = 0 \frac{dy}{dt} = 0 The Attempt at a Solution Here's what I did: 1 - xy= 0 ... x - y^3 = 0 1 - xy= 0 ... x = y^3 1 - y^4= 0 ... x =...

Homework Statement Find all the critical points of f(x,y) = 2x^3+xy^2+5x^2+y^2+100 Homework Equations The Attempt at a Solution I'm really not sure how to do this question due the the x^3 term in the function. Could someone please advise how to start this. Thanx :D

Homework Statement f(x,y,z)=(xy+yz+xz)/(1+x^2+y^2+z^2) Explain why f has no absolute maximum or minimum. How about critical points? Homework Equations Hint: it is simplest to make 3 cases: a) x+y+z does not =0 b) x+y+z=0 c) x=y=z=0 The Attempt at a Solution I did cases b and c...

For f(x,y) = x^2 + y^2 + 3xy I need to find the critical points and prove whether or not they are local minima, maxima or saddle points. I thought the only critical point was (0,0) since Df = (2x + 3y, 2y + 3x) = 0. Doesn't this make (0,0) a local min? The reason I doubt this now is because upon...

Hi, I posted a related question (about the 2nd derivative test) a while ago but now that I've read through some relevant theory I'm in a better position to find out about the following. In the notes I have, the behaviour of a function f:U \subset R^n \to R near a critical point point...

How would you go about calculating the volume, pressure or temperature for the critical point in a phase plane? I know that there's a Clapeyron equation for finding the equation of the coexistance curve dP= L/TV dT, but can this be used to find the critical point? and if not, what will? I've...

"Let a,b be in R with a>0 and f(x)=ax^3+bx. Let k(x)=[f''(x)]/[1+(f'(x))^2]^(3/2). Find the critical points of k(x) and use the first derivative test to classify them." This seems incredibly quantitative and complicated for an analysis assignment. There must be a theorem of some kind I can...

Given n\geq 2, n\in \aleph and f(x,y)=a*x^n+c*y^n where a*c\not=0, determine the nature of the critical points. I found the only critical point at (0,0) and when i tried to use the criteria of the determinant of the hessian matrix to determine the kind of critical point it was, it gave me 0, so...

Ok so i got a mathematica assignment that asks "Find and classify the critical points of f(x,y)= 5-10xy-4x^2+3y-y^4 What does this mean lol. The assignment was from 3 weeks ago and i just had spring break so my brain is completely outa wack. Does this mean classify the points as concave up...

How do I find the Critical points of a two-variable function using MATlab? I have a problem, I cannot seem to find the critical points of a two-variable function for the life of me! The funtion f(x,y) = 10x^2y - 5x^2 - 4y^2 - x^4 -2y^4 is supposed to have six potential critical points. I...

hi guys I'm stuck on this, i keep thinking i have the right answeres but i can't get them. the function is f(x): -2x^3 + 39x^2 - 180x + 1 i need to list all of the critical points, and hten indicate where it is increasing and where it is decreasing. I set the derivative -6x^2 + 78x - 180...

Intrinsic critical points of a function... I have a problem that's giving me a bit of trouble. The solution is not vital to my existence, but it's been eluding me long enough that it's grown to be a pest. I've figured out the easy part by hand, and I have the less obvious solutions by way of...

Say I have a function defined on all of \mathbb{R}^2 which is continuous everywhere, and of class C^{\infty}. To find the critical points, I simply find the points where the Jacobian is zero, right (since every point in the domain is in the interior of the domain). Then, to classify the...

point of inflection -- always halfway between 2 critical points?? when you look at the graph of a function say, f(x) = 5x^3 - 2x^2 + 3x - 1 will the point(s) of inflection always be equidistant from 2 critical points (ie the 2 nearest critical points)? point of inflection -- point...