Critical points Definition and 144 Threads

Homework Statement Locate and classify the critical points of f(x,y) = (x-y)(xy-1). Homework Equations The Attempt at a Solution I found the partial derivatives with respect to x and y and I got: ∂f/∂x = -y2+2xy-1 ∂f/∂y = x2-2xy+1 After setting them both equal to zero I can't...

Homework Statement I am trying to find the critical points of the following hyperbolic function: f(x) = a / (b + x) Homework Equations Critical points--> where f '(x) = 0 One of the points on the graph is a/2b The Attempt at a Solution I am not sure how to proceed with this...

Homework Statement f(x,y)= 16(y^2) +(x^4) y + 4(x^2) + 4 My problem is recognizing which critical points to consider valuables. Homework Equations fxx, fyy, fxy, and second partials test. D=fxx(fyy)- (fxy)^2 The Attempt at a Solution I found: fx=0 4(x^3)y +8x=0 (x^2) y= -2...

Critical Points, intervals, local max/min help! Calculus. 1. I need help with a homework problem that I just cannot get right. It asks: Answer the following questions about the functions whos derivative is given below. f'(x) = (sinx +1)(2cosx +\sqrt{3} ), 0\leqx\leq2∏ a. what are the...

Homework Statement Hi I have an equation as follows: f(x) = (2x-2.3)/(2x-5.29)^2 what i got for the derivative was: f'(x) = (-1.38-4x)/(2x-5.29)^3 Homework Equations f(x) = (2x-2.3)/(2x-5.29)^2 f'(x) = (-1.38-4x)/(2x-5.29)^3 The Attempt at a Solution what i got for the...

How would one show that if there is a number c for which g'(c)=0, then every point on the level set {(x,y)|H(x,y)=c} is a degenerate critical point of f? I know that the question may seem vague, but this is the question as it was given to me by my professor. It is something to think about...

Hi guys, I have questions about algebraic functions and not sure where to ask. Hope it's ok here. Given the algebraic function f(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0 I recall seeing a reference that stated as n increases, the critical points of the function migrate to the unit circle or...

Hello, I had a lecture in an elementary Calculus course today, and the professor used a method for classifying critical points which I can't see in my book (Calculus, a complete course, by Adams). Basically he had the function f(x,y)=x^2+2y^2-4x+4y. He found a critical point at (2,-1) -...

Homework Statement Locate the critical point of the function, e^-(x^(2)+y^(2)+2x) Homework Equations none The Attempt at a Solution Ok first step differentiate the function and set it to zero for both fx and fy, Fx(x,y) = (-2x-2)e^(-x^(2)-y^(2)-2x) Fy =...

Homework Statement g(x)=x - sin (pi*x) at [0,2] Homework Equations all values where g'(x)= 0 (or non-existant) are critical points The Attempt at a Solution g'(x)=1 - ╥(cos(╥X)) 1/╥=cos(╥X) cos^(-1) (1/╥) = ╥X X=(cos^(-1) (1/╥))/╥ x=.397 and x=? 2nd point

Homework Statement http://i.minus.com/jCH20SF290QIb.png Homework Equations Critical point: when the derivative = 0 or the derivative fails to exist. The Attempt at a Solution I got x = 0 and x = e as critical points. When x = e, the function becomes 0 / e, which = 0...

Homework Statement Find all the critical points of the system of DE: x'=x-x^2-2xy and y'=2y-2y^2-3xy.Homework Equations y'=f(x, \vec y) has critical point if f(\vec x )=0.The Attempt at a Solution I'm self studying DE's and I'm not understanding well what critical points are. It looks like I...

Homework Statement I need to find the critical points of this functionhttp://www4a.wolframalpha.com/Calculate/MSP/MSP84719i6a1iabg11598d000049hhhie62iif1c81?MSPStoreType=image/gif&s=16&w=140&h=46 The Attempt at a Solution I'm having trouble solving the first derivative, which is...

Homework Statement A perfectly smooth two-dimensional surface that has the shape of a doughnut is suspended in the (x, y, z)-Cartesian space. Let the doughnut be positioned so that an observer facing the plane y=0 will see through the hole. For example, the doughnut is standing up vertically...

Homework Statement I have been given a 4x4 hessian matrix at a point c. I want to find out if the point is a local min/max/saddle. I am aware of the conditions of critical points for 3x3 hessians, but I'm not sure about 4x4 hessians. The hessian is given by 1 0 -2 1 0 2 0 -1 -2 0 5 2 1 -1 2...

Homework Statement \ Find the critical points of the function. Then use the Second Derivative Test to classify each critical point (or state that the test fails). f(x, y) = x3 + y4 - 21x - 18y2 The Attempt at a Solution partial x derivative=3x^2-21=0 partial y= 4y^3-36y=0...

I am asked to find all local maximum and minimum points for the function, f(x,y) = (1+xy)(x+y) so, naturally, I had to find the critical points. However, in the back of my book (Calculus Early Transcendentals 6E, James Stewart) the only critical points are (1,-1) (-1,1) (both of which are...

Homework Statement Find all critical points and derive the linearised system about each critical point \dot{y} = [3y1 + y1y2; y1 + y2 - y22] Homework Equations The Attempt at a Solution Setting 0 = 3y1 + y1y2 so y2 = -3 0 = y1 + y2 - y22 so y1 = 12 = [0, 1; 12, -3] so...

Hi, I am studying for my Analysis final and came across this problem I just can't get my head around: Homework Statement Find the critical points of f(x,y) = (x^2+y^2-4)(x+y) and their nature. Homework Equations \vec{\nabla} f(x,y) = \vec{0} The Attempt at a Solution \frac{\partial...

Homework Statement I SOLVED THE MISTAKE WITH MY DERIVATIVE IN POST BELOW I never understood why we just get rid of the imaginary solutions? like for example if f(x) = (5*x)/(9-x^2) then df(x)/dx=(5[9+x^2])/(9-x^2) I've realized that when we set the numerator equal to zero we are doing this...

Trig function:-sinx+(1/2)sin2x(2) The criticial points are (pi), (pi/3), and (5pi/3). Am I correct?

Homework Statement I'm supposed to find (and classify, which I'm really not sure what that means) critical points of f(x,y) = x2y- x2 - 2y2 - 2x4 - 3y4 Homework Equations The Attempt at a Solution I found the partial derivatives, fx(x,y) = 2xy - 2x - 8x3 and fy(x,y) = x2 - 4y -...

In a RG flow diagram for a physical system where we have two fix points, which one is the most important one and describes the physical theory of the system? Is it the unstable fix point or the "stable" one where some of the flow lines flows towards?

Homework Statement g(x) = 4x - tanx Homework Equations The Attempt at a Solution g(\theta) = 4\theta - tan\theta g'(\theta)=4-sec^{2}\theta g'(\theta)=\frac{4cos^{2}\theta-1}{cos^{2}\theta} g'(\theta)=\frac{-4sin^{2}\theta}{cos^{2}\theta} -4tan^{2}\theta=0 -4(tan\theta)(tan\theta)=0...

Homework Statement I am trying to find the critical points of the system x1=x2+y2-2xy-1 and y1=y+x-2 Homework Equations x1=x2+y2-2xy-1=0 y1=y+x-2=0 The Attempt at a Solution I tried to solve for zero but I end up with multiple solutions way more then I should need any help answering...

Hey guys, can you just check and tell me if I took the derivative and found the critical points of the function correctly? It strikes as though I've made a mistake somewhere. http://img405.imageshack.us/img405/4098/funccrit.jpg

Homework Statement The function is defined in the xy-plane. Find all critical points and determine their characteristics. Homework Equations f(x, y) = x4 + 4xy + y4 The Attempt at a Solution I tried finding the partial derivatives. f'x = 4x3 - 4y f'y = 4y3 - 4x Setting them to 0...

I have a function which is: y = 27x2 + 2100/x dy/dx = 54x - 2100/x2 Critical points are for what x values, is y undefined or equal to 0? Therefore, in this case i can only see the value of 0, which would cause the function to be undefined. Due to x2 being on the denominator. EDIT...

Homework Statement I was given the following level curve (image is attached). I need to find the critical points and classify them (local max, min, saddle). The Attempt at a Solution I can find the points easily enough. Saddle at (-1,0), and then locals at each of the three...

Stability and Critical points---Please Help Homework Statement I need to know the critical points and stability of this ode (2xy-5)dx+(x^2+y^2)dy=0 Homework Equations The Attempt at a Solution i've graphed this is Mathematica (attached) but I don't know if its stable or what the...

Homework Statement Find the critical points of x3 + y3 + 3x2 + 6y2 - 9x + 9y +1 you do not need to define the critical points Homework Equations The Attempt at a Solution i have df/dx = 3x2 + 6x - 9 and when i solve this x = -3, 1 but i don't know what the corresponding...

i am told to investagate the nature of the critical points of the system: x'=e^y y'=(e^y)*cos(x) i am not sure where to begin because x' is always non-zero.

I'm trying to find the minimia and maxima of the following function without using LaGrange multipliers: f(x,y)=sin(x)+sin(y)+sin(x+y) where: 0\leq x \leq 2\Pi 0\leq y \leq 2\Pi Partial derivatives: f_{x}=cos(y)+cos(x+y) f_{y}=cos(x)+cos(x+y) f_{xx}=-sin(x+y) f_{xy}=-sin(y)-sin(x+y)...

Homework Statement Find and classify all critical points of, f(x,y)= x^{3} - 6xy + y^{3} Homework Equations The Attempt at a Solution See figure attached for my attempt at the solving all the critical points. I'm getting some weird numbers so I want to make sure I did this...

Hey, I have trouble finding the critical points for this function: z = f(s,t) = (1+s)(1+t)(s+t) I get that s' = (1+t) (t+2s+1) = 0 t' = (1+s) (s+2t+1) = 0 So t= -1 and s= -1 is a C.P. How do i solve for the others? Any help is greatly appreciated thanks!

In local coordinates what does it mean for a critical point of a function f:M\rightarrowR to be non degenerate? In addition how can you show that the definition is independent of the choice of local coordinates? I know that being a critical point is independent of the choice of local...

Homework Statement a)f(x) = sqrt(3 + x^2) - 3 x, text( ) text( ) [4, 6] [Find the max. and min. values] b)y = (3 - x)/(x^2 + 9 x), [2, 7] The Attempt at a Solution a)I know how to get the max. and min. value, but before knowing them I need to know how to get the critical points from f(x)...

Homework Statement Find the critical points and use the second derivative test to decide if your critical points are local maxima, local minima, or saddle points. f(x,y)=x4+y4+4xy The Attempt at a Solution so I took the gradient to get: <4x3+4y, 4y3+4x> I know I need to set...

1. Find the minimum point of I(p, q, r, s)=\int_{\mathbb{R}^2}\,dxdy\,\left[exp(-(x-d)^2-y^2)+exp(-(x+d)^2-y^2)-exp(-(x-p)^2-(y-q)^2)-exp(-(x-r)^2+(y-s)^2)\right]^2 +\int_{\mathbb{R}^2}\,dxdy\,\left[exp(-(x-\delta_x)^2-(y-\delta_y)^2)-exp(-(x-p)^2+(y-q)^2)\right]^2 for given d, \delta_x...

I'm having some trouble classifying critical points when the 2nd derivative test fails. The function is, f(x,y) = x^4 - 3x^2y^2 + y^4 The only critical point I've found is (0,0). How do I know that this is the only possible critical point? Anyways, the second derivative test fails...

Homework Statement Find all critical points of the function given.(see figure) Choose and classify anyone critical point. Homework Equations The Attempt at a Solution I found the following critical points, (0,0), (0,-3), (3/2,3/2) and (-3/2, 3/2) Wolfram alpha is...

Critical Points and Differential Equations! Helppp Hello everyone..Find it embarrassing enough on asking a question on my very first post but I've been an avid reader of the forums for the past couple of months and been finding what i need for all my assignments here.So a big Thank You to all...

Homework Statement find the derivative and critical points of the function f(x)= x^3-x^2+5 Homework Equations The Attempt at a Solution I tried to find the derivative: 3x^2-2x but I am not sure how to find the critical points from here...

Critical points of maps f:R^n-->R^m , m>1 Hi, Everyone: I am having trouble figuring out what the critical points would be for any such map f:R^n -->R^m , m>1. Problem is that I know sufficient conditions for differentiability (partials exist and are continuous.) but...

Homework Statement I don't have a particular problem in mind, just trying to recap a method my professor quickly skimmed over for determining whether a point is a max or a min. I had suggested that we use the second derivative test on our critical to determine whether it is a max or a min...

Homework Statement I don't believe that it's possible for a function to not have a local maxes and minimums. I have always been told that if you set the first derivative of a function to zero and solve for x and get imaginary numbers for your critical values that there are none. Now I don't...

why is it that in algebra 2 people are told about complex numbers then they magically disappear for like 5 years and there not suppose to exist... sorry for wrong title I can't change it =(

Homework Statement Determine the critical points f(x,y) = x^3y + xy Homework Equations The Attempt at a Solution fx(x,y) = 3x^2 + y 3x^2y+y=0 y(3x^2 +1) = 0 y = 0 3x^2=1 x=1/3fy(x,y) = x^3 + x x^3 + x = 0 x(x^2+1) = 0 x=0 x^2= -1 x= -1 I don't know how to take it from here as I have...

Hello. I'm studying for an ODE/PDE qualifier and I'm wondering how to do this problem. I feel like it should be pretty easy, but anyway.. Show that a Hamiltonian system in \mathbb{R}^{2n} has no asymptotically stable critical points. Any suggestions? Thanks..

Homework Statement Ok so I was trying to find the range of the equation below, commen sense and simply looking at it yields that the range is set of all reals... I however used calculus to find the critical points, I've always been told that there are none when the range is the set of all...