What is Critical points: Definition and 148 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. Z

    Why can't a critical point of a system of DEs be complex?

    From the first equation we can write $$y=\frac{x}{2}+\frac{x^2}{8}$$ Subbing into the rhs of the second equation and equating to zero we find (after some algebra) that $$x(x-4)(x^2+12x+72)=0$$ This equation has roots ##0##, ##4##, and ##-6\pm 6i##. Then, ##x=0\implies y=0## and...
  2. Poetria

    Critical points of a multivariable function (wave equation)

    I would like to check my understanding of this problem. There are the following possibilities: a. Isolated points where the gradient is 0. b. The level curves of height 0 c. The level curves of height 1. d. The level curves of height -1. e. None of the above. I would choose a, c, d. Where...
  3. mcastillo356

    Calculate (and argue) the critical points of an exponential function

    Hi everybody We can't differentiate ##x^x## neither like a power function nor an exponential function. But ##x^x=e^{x\mbox{ln}x}##. So ##\dfrac{d}{dx}x^x=\dfrac{d}{dx}e^{x\mbox{ln}x}=x^x(\mbox{ln}x+1)## And here comes the doubt: prove the domain of ##x^x## is ##(0, +\infty)## Why is only...
  4. S

    B Critical points of second derivative

    If the sign on the sign diagram of f" changes from positive to negative or from negative to positive, this means the critical points of f" is non-horizontal inflection of f But what about if the sign does not change? Let say f"(x) = 0 when ##x = a## and from sign diagram of f", the sign on the...
  5. D

    Critical points and partial differentiation

    zx = 2xy + y2 -3y = 0 and zy = 2xy + x2 - 3x = 0 Subtracting one equation from the other gives y2 - 3y = x2- 3x ⇒ y (y-3) = x (x-3) This leads to the following solutions ( 0 , 0) , (0 ,3) , (3 , 0) but the answer also gives ( 1, 1) as a solution. What have i done wrong to not get this...
  6. C

    I An interesting point regarding critical points and extrema

    Hi all, I have recently faced some problem about distances between two curves, and (re?)"discovered" an interesting point that I would like to share with you. In the following, we consider a function of two variables ##f(x,y)##, but it should be clear that the definitions and the result is...
  7. G

    MHB Are Critical Points and Roots Interchangeable in Math?

    Might be I am asking a silly question but really want to clarify that would critical points and roots are same terms use interchangeably? I mean we can use critical point as value of x and root also as value of x then what is the difference between?
  8. D

    Odd/even function and critical points

    I have ##3x^{2/3}## as an even function although there is some debate as to this in another thread I started but the (5-x) factor means the function is neither odd or even. I also see the domain as all real numbers. Hopefully this is right ? To find the critical points I differentiate f(x) to...
  9. Kenneth1997

    Directional derivatives at critical points

    x*abs(y)*(y+x^2+x)=f(x,y) so, on normal points they are tangent vectors on some point in the chosen direction. how about in critical points, where there shouldn't be any on a geometrical standpoint? can i say they exist if i get them with the definition? or the result i get has no value? like...
  10. Phys pilot

    Critical points of a diff eq system

    Homework Statement I have to calculate the critical points of the following system. $$x'=cx+10x^2$$ $$y'=x-2y$$ The Attempt at a Solution So I solve the system $$cx+10x^2=0$$ $$x-2y=0$$ So if $$x=2y$$ I have $$2yc+10*4y^2=2yc+40y^2=y(2c+40y)=0$$ and I get $$y=0$$ and $$y=-\frac{c}{20}$$ f I...
  11. Peter Alexander

    Critical Points of a Parameter Dependent Integral

    1. The problem statement, all variables, and given/known data Find and categorize extremes of the following function: $$F(y)=\int_{y}^{y^{2}}\frac{1}{\ln^{2}x}dx$$ for ##y>1##. Homework Equations $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...
  12. R

    I Critical points for difficult function

    Given that f'(x)=[(8cos(x)/(x2)-(⅛)], find the number of relative maxima and minima on the interval (1,10). Finding maxima and minima analytically wasn't fruitful for me, so instead I used a bit of handwaving. First I argued, using the intermediate value theorem, since f'(1)>f'(10), there...
  13. M

    MHB Are the critical points minima or maxima?

    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...
  14. harpazo

    MHB Critical Points & Extrema of Multivariable Function

    Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. f (x, y) = x^(2/3) + y^(2/3) Solution: f_x = 2/[3 (x)^1/3] f_y = 2/[3 (y)^1/3] f_xx = -2/[9 x^(4/3)] f_yy = -2/[9 y^(4/3)] f_xy = 0 I set f_x and f_y to 0 and found...
  15. harpazo

    MHB Critical Points & Extrema of f (x, y)

    Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. f (x, y) = x^3 + y^3 Solution: f_x = 3x^2 f_y = 3y^2 f_xx = 6x f_th = 6y f_2xy = 0 I set f_x and f_y = 0 and found the critical points to be (0, 0). Is this...
  16. harpazo

    MHB Finding Critical Points and Extrema for g(x, y) = sqrt{x^2 + y^2 + 1}

    Find the critical points and extrema of the function g (x, y) = sqrt {x^2 + y^2 + 1}. Can someone get me started here? I also would like the solution steps. I said solution steps not the solution. Do it like this: Step 1... Step 2... Step 3...etc...
  17. C

    Solving System of Equations - Critical Points

    Homework Statement [/B] Hi, I seem to have this issue with solving the a system of 3 equations when trying to find the critical points for a 3 variable function. Homework EquationsThe Attempt at a Solution It always keeps happening. I keep getting stuck when solving these equations. Every...
  18. Cjosh

    Pulling fractional exponents out of an expression

    Homework Statement Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4 Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4 Homework EquationsThe Attempt at a Solution I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...
  19. mr.tea

    I Lagrange multipliers and critical points

    Hi, I have (probably) a fundamental problem understanding something related critical points and Lagrange multipliers. As we know, if a function assumes an extreme value in an interior point of some open set, then the gradient of the function is 0. Now, when dealing with constraint...
  20. I

    MHB Finding Solutions for $y'=Ay$ with Vector and Matrix Components

    I think this should be my last question :D It's a question i missed on my last exam and have no idea how to do it and I wanted to quickly go over because my final is in a few hours so if anyone could help that would be awesome View image: 20151215 092327
  21. I

    MHB What are the different types and stabilities of critical points?

    identify the type of critical point as spiral, center, saddle, proper node or improper node. in addition, identify the stability of the critical point as stable, unstable, or attractive and stable. View image: 20151215 084536 1 In order: proper node, saddle point, improper node, center But I...
  22. N

    Finding critical points for phase planes

    Homework Statement X'=x-y^2 Y'=x^2 -xy -2xHomework Equations Find the critical point of this system. The Attempt at a Solution I know one point is (0,0). When I tried it I got sqrt(x)= 2 and sqrt(x)= 1 but the point (1,1) doesn't fit later parts of the question. I'm also aware that since a...
  23. qq545282501

    How do I find critical points and determine local extrema for a given function?

    Homework Statement find all critical points and identify the locations of local maximums, minimums and saddle points of the function f(x,y)=4xy-\frac{x^4}{2}-y^2 Homework EquationsThe Attempt at a Solution setting Partial derivative respect to x = 0 : 4y-\frac{4x^3}{2}=0 setting partial...
  24. N

    Critical Points of a Function: Finding the Points of Inflection

    Homework Statement 1.) Let g(x) = x − (5/x^2) find all critical numbers (if any) of g. Give answers in increasing order (smallest first). Enter DNE in any unused space. x = x = and 2.) Let g(x) = x |x + 5| The Attempt at a Solution [/b] I know I must first take the derivative and then...
  25. C

    Critical points and min and max

    Homework Statement Hi I'm suppose to find the critical point and minimun or maximum of f(x)=(4/5) Homework Equations [/B] I have a question regarding how to interprete the results The Attempt at a Solution1) we start by finding f'(x)=4/(5*x^1/5) Now my first question is this: we cannot...
  26. C

    Critical points and of polynomial functions

    Homework Statement A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the...
  27. C

    Intervals of increase and decrease

    Homework Statement Find the intervals of increase/decrease. Use first derivative test to find the local maxima and minima. Sketch a rough graph. Homework Equations a) f(x) = 2x2+12x-1 b) f(x) = 1/2x4-2x2 c) f(x) = 3x-4 sqrt 2 d) f(x) = x2-1/x2+1 The Attempt at a Solution A) it becomes f...
  28. Khronos

    Optimisation - Critical Numbers for Complex Functions.

    Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows: Question: Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula: V = 999.87 −...
  29. A

    MHB Critical Points and Global Extrema Question

    Hey guys, I'm doubting some of my answers and I'd appreciate some help. I'm only asking about 2abc, ignore 1ab please: For 1a, I simply took the derivative (as I did with all three of these questions) and calculated global and local extrema and critical points. Ultimately, I found that...
  30. Q

    Understanding Phase Diagrams: Boiling, Melting & Critical Points

    Homework Statement https://scontent-b-dfw.xx.fbcdn.net/hphotos-prn1/v/1012733_10201076259690370_236735593_n.jpg?oh=4896ca67a34d3b223b764aba8ed308ab&oe=528A1B57 Homework Equations http://upload.wikimedia.org/wikipedia/commons/3/34/Phase-diag2.svg The Attempt at a Solution Not sure entirely...
  31. A

    MATLAB Critical points of system of ODE in MATLAB - Game theory and poker

    Hello there, I hope I'm posting in the right section. I have been doing some work on evolutionary game theory and poker. I will give a brief description of how I got here. I have eight strategies i = 1, 2, \ldots, 8 and the eight proportions of the population playing each strategy is...
  32. C

    How to find critical points for a differential equation?

    Homework Statement f(x,y)= 3x5y2 - 30x3y2 + 60xy2 +150 How do I find the critical points where this equation equals zero? Homework Equations None? The Attempt at a Solution I found the partial derivatives, but I'm stuck now and don't know where to go from here. dx = 15x4y2...
  33. J

    Find all critical points, and identify them as minima, maxima, or

    Homework Statement Consider the function f(x,y) = (x2 + 4y2)e(1-x2-y2) Find all critical points, and identify them as maxima, minima, or saddle points. The Attempt at a Solution I took the partial of x and the partial of y, and set them equal to 0. This is what I got: fx(x,y) =...
  34. Ascendant78

    How to find derivative critical points that DON'T use ref. angles?

    Homework Statement I am trying to figure out how to solve for problems that give a function and ask for the critical points. However, the ones I am having difficulty with are the ones that have multiple trigonometric functions included in them.Homework Equations Find the critical points of the...
  35. D

    Find all the critical points of the function

    Homework Statement f(x) = x ln(x) The attempt at a solution f'(x) = product rule, resulting in 1 + ln (x) So by way of solving the problem, set 1 + ln (x) = 0 Now idealistically, find something in x that, when added to 1, equals zero. Now here's a problem I have. How do I know when to...
  36. N

    Weird theorem on critical points for multivariable functions

    Why does it write all that stuff about fxy2? Isn't it unnecessary? I mean, isn't the following true? If in a point (a,b): fx=fy=0 and fxxfyy>0, then the partial derivatives must be either both negative or positive, and thus point (a,b) is a local minima or maxima. And if fxxfyy<0 => (a,b) is a...
  37. T

    Critical points (x-1)^4 (x-y)^4

    Homework Statement Find critical points. f(x,y) = (x-1)^4 + (x-y)^4 The Attempt at a Solution \frac{\partial f}{\partial x} = 4(x-1)^3 + 4(x-y)^3 \frac{\partial f}{\partial y} = - 4(x-y)^3 \frac{\partial f}{\partial x} = 0 , \frac{\partial f}{\partial y} = 0 , -...
  38. R

    Critical Points of cos y & cos 2x - Why the Difference?

    for cos y the critical points are n*pi - 0.5*pi for cos 2x, the critical points should be n/2 * pi - 0.25*pi because if I set y=2x = n*pi - 0.5*pi , I get the eq in red for x. However if I do it graphically, I just get n*pi - 0.25*pi. The question: why algebraically I am constrained to use...
  39. B

    Graphing a function using critical points and increasing/decreasing intervals

    Homework Statement Find the local maxima and minima and sketch: Critical points: (3, -4) and (6, 0) Interval of increase: (3, ∞) Interval of decrease (-∞, 3) I'm not quite sure if I graphed this correctly since I wasn't given the function to doublecheck. The Attempt at a...
  40. L

    Classify the Critical Points (Advanced)

    Good Day, My professor historically has put more advanced questions on the final exams than can be found in our textbook. My supplements and extensive search engine use have not allowed me to get any further...Ive hunted through as many examples as I could find but cannot seem to find a...
  41. T

    MHB Finding Critical Points of a Dynamical System.

    Hello guys . I obtained a 3 dimensional dynamical system , how can I find its critical points with using software ? I tried it handy but its too involved to compute handy .
  42. B

    Help with finding first derivative and critical points?

    Homework Statement What are the critical points of function f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 ?Homework Equations When f'(x) equals 0 or is undefined, x is a critical number.The Attempt at a Solution The original function is f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 . I got the derivative as f'(x) =...
  43. D

    Are All Inflection Points Also Critical Points?

    Are inflection points critical points? and what about at the value that f(x) undefined? Is that critical point too?
  44. W

    Critical Points & their Nature of a Multivariable Function

    Homework Statement f(x,y) = xy(9x^2 + 3y^2 -16) Find the critical points of the function and their nature (local maximum, local minimum or saddle) Homework Equations The Attempt at a Solution I have partially differentiated the equation into: fx = 27yx^2* + 3y^3 -16y fy =...
  45. N

    Critical Points and Graphs of Differential Equations

    Consider the equation dy/dt = alpha - y^2 a) Find all of the critical points. How does it change as alpha < 0, alpha = 0 or alpha > 0? b) In each case of different alphas, consider the graph of f(y) vs y and determine whether each critical point is asympototically stable, semistable, or...
  46. F

    Translating a linear system to have critical points at the origin?

    So suppose you have a nice simple linear system dx/dt = a1x + b1y + c1 dy/dt = a2x + b2y + c2 and a critical point (x0, y0) ≠ (0,0). If you want to analyze the stability in the usual linear method, then I assume there should be some transformation to have the critical point at the origin. So is...
  47. S

    Critical points of differential equation

    Homework Statement Determine the location and type of all critical points of the given equations and sketch the phase portrait y"+cosy=0 The Attempt at a Solution I've done some like this before but they were all systems of equations. I'm actually not sure how to do the...
  48. T

    Quick question about use of the Hessain for critical points

    So I am solving for the critical points of x3-3x-y2+9y+z2 I've found the critical points and I'm just evaluating them in the Hessain. Now I computed the eigenvalues for the hessian, but one of my 2nd order derivatives was a constant. So my hessian looked like this (it was diagonal, so I'm...
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