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.
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...
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...
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 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...