Critical points Definition and 144 Threads

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...

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...

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...

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...

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...

Is my work right?

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?

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...

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...

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...

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 $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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 −...

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...

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...

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...

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) =...

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...

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...

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...

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 , -...

Jordan writes:

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...

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...

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...

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 .

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) =...

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

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 =...

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...

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...

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...

I have the equation (x^2)/(sqrt(x+1)) This one has been stumping me, not sure how to reduce the derivative properly much less solve for 0. I get it down to this using the quotient rule: ((x+1)^(1/2)*2x - x^2*1/2(x+1)^(-1/2)) / (x+1) Just started learning derivative a few weeks ago...

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...

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...