What is Critical points: Definition and 147 Discussions

Hazard analysis and critical control points, or HACCP (), is a systematic preventive approach to food safety from biological, chemical, physical hazards and more recently radiological hazards in production processes that can cause the finished product to be unsafe and designs measures to reduce these risks to a safe level. In this manner, HACCP attempts to avoid hazards rather than attempting to inspect finished products for the effects of those hazards. The HACCP system can be used at all stages of a food chain, from food production and preparation processes including packaging, distribution, etc. The Food and Drug Administration (FDA) and the United States Department of Agriculture (USDA) require mandatory HACCP programs for juice and meat as an effective approach to food safety and protecting public health. Meat HACCP systems are regulated by the USDA, while seafood and juice are regulated by the FDA. All other food companies in the United States that are required to register with the FDA under the Public Health Security and Bioterrorism Preparedness and Response Act of 2002, as well as firms outside the US that export food to the US, are transitioning to mandatory hazard analysis and risk-based preventive controls (HARPC) plans.It is believed to stem from a production process monitoring used during World War II because traditional "end of the pipe" testing on artillery shells' firing mechanisms could not be performed, and a large percentage of the artillery shells made at the time were either duds or misfiring. HACCP itself was conceived in the 1960s when the US National Aeronautics and Space Administration (NASA) asked Pillsbury to design and manufacture the first foods for space flights. Since then, HACCP has been recognized internationally as a logical tool for adapting traditional inspection methods to a modern, science-based, food safety system. Based on risk-assessment, HACCP plans allow both industry and government to allocate their resources efficiently by establishing and auditing safe food production practices. In 1994, the organization International HACCP Alliance was established, initially to assist the US meat and poultry industries with implementing HACCP. As of 2007, its membership spread over other professional and industrial areas.HACCP has been increasingly applied to industries other than food, such as cosmetics and pharmaceuticals. This method, which in effect seeks to plan out unsafe practices based on science, differs from traditional "produce and sort" quality control methods that do nothing to prevent hazards from occurring and must identify them at the end of the process. HACCP is focused only on the health safety issues of a product and not the quality of the product, yet HACCP principles are the basis of most food quality and safety assurance systems. In the United States, HACCP compliance is regulated by 21 CFR part 120 and 123. Similarly, FAO and WHO published a guideline for all governments to handle the issue in small and less developed food businesses.

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  1. fluidistic

    Stability of 2 critical points in a system of DE's

    Homework Statement Basically I found the following system of DE's: \frac{dx}{dt}=y \frac{dy}{dt}=-\frac{g}{l} \sin x - \frac{cy}{ml}. (Damped pendulum) I'm asked to analize the stability of the critical points x=0, y=0 and x=\pi, y=0. Using intuition the first point is asymptotically stable...
  2. I

    Critical points of multivariable equations

    Homework Statement Find all critical points of u(x,y)=(x-y)(x^2+y^2-1) Homework Equations - The Attempt at a Solution Partial differentials: ux=3x^2+y^2-1-2xy=0 uy=-x^2-3y^2+2xy+1=0 I know the critical points are the solutions to the above two equations. But how do...
  3. H

    Critical points with multiple variables

    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...
  4. T

    Critical points of a Hyperbolic function

    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...
  5. K

    Find and Classify the critical points of f(x,y)

    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...
  6. N

    Critical Points, intervals, local max/min help Calculus.

    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...
  7. S

    Finding increasing/decreasing intervals of an equation using critical points?

    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...
  8. A

    Level Sets and Degenerate Critical Points

    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...
  9. J

    Limit of critical points of algebraic functions

    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...
  10. R

    Critical points (will be a function of beta and delta)

    Homework Statement Find the critical points where \alpha = 1. dv/dt = v2 + \alphav - u + \delta [I will call this (1)] du/dt = \betav - u [call this (2)] For what values of \beta and \delta are there no critical points? Homework Equations The Attempt at a Solution So...
  11. H

    How Are Critical Points Classified in Calculus?

    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) -...
  12. D

    Critical Point of Multivariable Function

    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 =...
  13. C

    Critical points of a trig function

    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
  14. Q

    Critical Points of Log Function

    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...
  15. fluidistic

    Critical points of system of DE

    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...
  16. A

    What are the critical points of a given function and how can they be solved?

    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...
  17. S

    Critical Points of Doughnut Surface

    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...
  18. A

    Critical points of 4 variables?

    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...
  19. K

    Finding the critical points of a multivariable function

    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...
  20. D

    Calculating Critical points for multivariable functions.

    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...
  21. R

    Solving an autonomous system finding critical points

    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...
  22. R

    Find the critical points of (x^2+y^2-4)(x+y) and their nature

    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...
  23. G

    Some questions about solving for critical points

    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...
  24. R

    Critical Points of a trig function

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

    Finding Critical Points of a multivariable function

    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 -...
  26. I

    Renormalisation group - Which critical points are usefull in a RG flow diagram?

    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?
  27. QuarkCharmer

    Finding Critical Points in g(x) = 4x - tanx

    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...
  28. D

    Finding critical points of a non-linear sysytem

    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...
  29. Femme_physics

    Taking the derivative and finding critical points did I get it right?

    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
  30. I

    How do I find the critical points of this multivariable function?

    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...
  31. R

    What Is the Exact Value of x That Minimizes y for Positive x?

    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...
  32. T

    Finding critical points from level curves

    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...
  33. P

    Stability and Critical points-

    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...
  34. S

    Critical points, several variables

    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...
  35. R

    Critical Points in a system of differential equations

    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.
  36. L

    Finding Critical Points Of A Function

    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)...
  37. jegues

    Critical Points of f(x,y)= x^{3} - 6xy + y^{3}

    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...
  38. T

    Finding critical points of functions of two variables

    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!
  39. P

    Non degeneracy of critical points

    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...
  40. A

    Finding Critical Points in Fractional Functions

    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)...
  41. S

    Critical Points and the Second Derivative Test for a Multivariable Function

    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...
  42. O

    Find critical points, please share any tricks you know of

    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...
  43. jegues

    Classifying Critical Points when the 2nd derivative test FAILS

    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...
  44. jegues

    Identifying and Classifying Critical Points

    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...
  45. V

    Critical Points and Differential Equations Helppp

    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...
  46. N

    How Do You Determine Critical Points from a Derivative?

    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...
  47. B

    Critical points of maps f:R^n->R^m , m>1

    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...
  48. jegues

    Determining Max/Mins, Critical Points (Conceptual)

    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...
  49. G

    I don't buy that it's possible for a function to not have critical points

    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...
  50. G

    Can a function really have no critical points?

    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 =(
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