# What is Critical points: Definition and 148 Discussions

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1. ### 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. ### 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. ### 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. ### 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. ### 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. ### Given ##f(x)=3x^5-5x^3##, find all critical points.

Is my work right?
7. ### 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...
8. ### 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?
9. ### 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...
10. ### 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...
11. ### 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...
12. ### 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 $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...
13. ### 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...
14. ### 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...
15. ### 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...
16. ### 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...
17. ### 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...
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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
22. ### 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...
23. ### 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...
24. ### 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...
25. ### 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...
26. ### 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...
27. ### 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...
28. ### 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...
29. ### 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 −...
30. ### 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...
31. ### 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...
32. ### 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...
33. ### 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...
34. ### 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) =...
35. ### 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...
36. ### 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...
37. ### 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...
38. ### 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 , -...
39. ### MHB Jordan's question from Facebook (finding critical points)

Jordan writes:
40. ### 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...
41. ### 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...
42. ### 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...
43. ### 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 .
44. ### 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) =...
45. ### 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?
46. ### 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 =...
47. ### 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...
48. ### 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...
49. ### 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...
50. ### 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...