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. S

    Find Critical Points of f(x,y)

    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...
  2. X

    Critical points of a Hamiltonian system

    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..
  3. G

    Range of cos(x)=-2: Exploring Critical Points

    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...
  4. K

    Critical points of function in 3 variables

    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.
  5. jegues

    Critical Points in the First Quadrant

    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...
  6. G

    Calculating critical points and classifying them

    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...
  7. L

    Construction of Reeb Graph for Critical Points on Contours

    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
  8. K

    Finding some critical points [basic algebra difficulty]

    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...
  9. K

    Find the critical points of this function:

    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...
  10. K

    How do you determine the behavior of critical points when you have the Hessian?

    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...
  11. X

    Analyzing Critical Points of f(x,y)=Ax2+E

    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...
  12. H

    Finding Critical Points of P: Sum Notation Explained

    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} -...
  13. M

    Critical Points - Multivariable Calc

    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...
  14. S

    How many critical points are there in a function with multiple variables?

    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...
  15. C

    Finding Critical Points for f(x, y) = sinx + siny + cos(x+y)

    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...
  16. L

    Critical Points of f(x)=\frac{\sqrt[3]{x-4}}{x-1} - Max/Min Value

    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}=...
  17. C

    Critical points and 2nd partials test

    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)...
  18. JJBladester

    Kirchoff's Law and Critical Points

    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) +...
  19. D

    Critical points of multivariables

    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...
  20. M

    What is the Mistake in Finding the Critical Point for f(x,y)=xy+(144/x)+(12/y)?

    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...
  21. J

    How to Find Critical Points of f(x) = xln(x)

    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...
  22. Saladsamurai

    LeGrange Multipliers Finding critical points of function

    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...
  23. Saladsamurai

    Critical points of 2-Variable Function

    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...
  24. Somefantastik

    Find 2 Missing Critical Points for x' = y(1 + x - y2) and y' = x(1 + y - x2)

    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 -...
  25. L

    Sketching curves: intercepts, asymptotes, critical points [answer check]

    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...
  26. A

    Three Critical Points and Type of Local Phaseportrait

    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...
  27. F

    Finding Critical Points: Solving f_x & f_y

    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...
  28. G

    Solving 2 Problems: Finding Critical Points & Differentiability

    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...
  29. G

    Classifying Critical Points of Multivariable Functions

    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...
  30. M

    What is the critical point of xe^x?

    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.
  31. R

    Find Critical Points of x' and y': ax - bxy & bxy - cy

    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...
  32. L

    Do we include asymptotes as critical points?

    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...
  33. W

    Determining critical points

    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...
  34. E

    Find Critical Points of dx/dt and dy/dt Equations

    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 =...
  35. C

    What are the critical points of a function with multiple variables?

    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
  36. E

    Understanding Critical Points in Multivariable Functions

    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...
  37. B

    Finding Critical Points and Local Extrema of a Multivariable Function

    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...
  38. B

    Critical points of functions

    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...
  39. M

    Thermodynamics: Critical points

    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...
  40. T

    Solving a Complex Analysis Problem: Finding Critical Points of k(x)

    "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...
  41. P

    Analyzing Critical Points of a Multivariable Function

    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...
  42. Pengwuino

    Calculus III: Confused about Critical Points of f(x,y)?

    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...
  43. V

    MATLAB How do I find the Critical points of a multi-variable function using MATlab?

    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...
  44. S

    Critical Points of f(x): -2x^3 + 39x^2 - 180x + 1

    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...
  45. S

    Intrinsic critical points of a function .

    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...
  46. A

    Quick Question about Critical Points

    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...
  47. brum

    Point of inflection - always halfway between 2 critical points?

    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...
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