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Questions tagged [stationary-point]

A stationary point is a point on a graph of a function where the derivative of the function is zero. This tag is for questions involving the existence and classification of stationary points. For questions focusing on minimizing of maximizing values under constraints, use (optimization).

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First & second derivative of $g(t)=f(\alpha+t\cos\theta,\beta+t\sin\theta)$ and conditions for minimum

If $f(x,y)$ is a smooth real-valued function, and $g(t)=f(\alpha+t\cos\theta, \beta+t\sin\theta)$ express $g’(t)$ and $g’’(t)$ in terms of partial derivatives of $f$. Then deduce the conditions for $t$...
edster101's user avatar
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Method of Steepest Descent (deform contours where there are 2 saddles)

Question: use the method of steepest descent to obtain the first two non-zero terms in the asymptotic approximation $$\int_0^\infty \exp(ix(t^3/3+t))dt\sim i(1/x+2/x^3+...+a_n/x^n)$$ as $x\to\infty$ ...
vegetandy's user avatar
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Classify stationary points of $f(x) = 2x_1^3 - 6x_2^2 + 3x_1 ^2x_2$

Given be the function $f(x) = 2x_1^3 - 6x_2^2 + 3x_1 ^2x_2$. I would like to compute and classify all stationary points $x^*$. \begin{align*} \text{Function:} \ f( x) & \begin{array}{l} =2x_{1}^{...
PatrickSteiner's user avatar
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How to use stationary phase method when the zero point is at infinity?

Thank you for reading my questions. It is known that the stationary phase method has this form: $$ \int_a^bg(t)e^{jf(t)dt}\approx\sum\limits_{t_0\in\Sigma}g(t_0)e^{jf(t_0)+j*\text{sign}(f''(t_0))\frac{...
Xiangyu Cui's user avatar
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58 views

Show that $\dot{x} = y, \dot{y} = -x +\frac{3}{2}x^2$ has a non-periodic, bounded solution using the Lyapunov-function.

As stated in the title I have the following system of ODE's $$\begin{aligned}\dot{x} &= y,\\ \dot{y} &= -x +\frac{3}{2}x^2\end{aligned}$$ I have already found the stationary points $(0,0)$ and ...
designerresearch44's user avatar
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What is the meaning of a stationary value in real life?

So I just solved a question saying that there is a sector with radius "$r$" and sector angle "$ \theta $" radians, the total area of the sector is "$A$" and the perimeter ...
tuna21's user avatar
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2 votes
2 answers
103 views

Determine stability of non-hyperbolic stationary point

Given the system $$\begin{align*} \dot{x_1} &= x_2+x_1^2-x_1^3 \\ \dot{x_2} &= -x_2+\mu x_1^2 \end{align*} $$ determine the stability of the stationary point in the origin for $\mu = \{-1,0, 1\...
Carl's user avatar
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Second derivative $= 0$ at turning point $(0,0)$ for $y = x^5-5x^4$

My understanding is that at a maximum turning point of a function the first derivative is zero and the second derivative is negative. However, I'm confused that with $y = x^5-5x^4$ the turning point $(...
Nimna De Silva's user avatar
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1 answer
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Classification of saddle points

PREMISES In my multivariable calculus book, a saddle point for a function $f: Dom(f) \subseteq \mathbb{R}^n \rightarrow \mathbb{R} $ is defined as a stationary point which is not a minimum or maximum. ...
selenio34's user avatar
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Method of Steepest Descent and Contour Deformation

In the book An Introduction to Quantum Field Theory by Peskin and Schroeder, p. 14 in section 2.1, it is stated that, in looking at the asymptotic behavior for $x^{2} \gg t^{2}$ of the integral \begin{...
Leonardo's user avatar
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To Find the extreme values of the function by Lagrange method of undetermined multipliers

Find extreme values of the function $f(x,y) = xy$ on the surface $g(x,y) = \frac {x^2}{8} + \frac {y^2}{2} - 1 = 0$. My approach : First I created an auxilliary function $F(x,y,λ) = xy + λ(\frac {x^2}{...
Subhash Kshatri's user avatar
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2 answers
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$f(x, y)=exp(-3x)+exp(-y)+5x^2y^2$ initial guess $x^0=0, y^0=0$

Qno: Find critical points for $f$. Can anyone help to understand this and how to solve it? Do I have to use first derivative test or is there any other numerical method to solve this type of problem ? ...
Waseem Bughio's user avatar
3 votes
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Asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$?

What are asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$ ? Here is the graph for $n=10$. The turning point in $x\in(0,1)$...
Dan's user avatar
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finding stationary solution of a continuous time markov chain.

With a certain rate $R$ balls fall into a box. There is no limit to the number of balls the box can hold, but each ball has a rate $\gamma$ to leave the box and when two balls hit each other they ...
bawo__'s user avatar
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Question on estimate in one of Jean Bourgain's 1992 papers

The goal of the argument is to estimate the following integral: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\varphi_k(|\xi|)^2\,d\xi$$ where $t(x):x\in B(0,1)\mapsto t(x)\in(0,...
Diffusion's user avatar
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Why is it justifiable to go into coordinates when performing calculus of variations on manifolds?

Pretty much every time I see someone write down an action and find it's stationary points they immediately switch to coordinates and expand the Euler-Lagrange equations. Specifically, take the ...
Chris's user avatar
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Clarification about maximum and minimum points

I would like some calrification about the question of max and min for a function in one variable. My doubts are these: our professor told us that when searching for max and min we have to pay ...
Heidegger's user avatar
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Number of stationary points of a Polynomial [duplicate]

Consider a polynomial with degree n. Then the greatest number of stationary points it may have is n-1. How can we build intuition or prove for why this is the case? However, my main question is as ...
James Chadwick's user avatar
1 vote
1 answer
70 views

Clarification about inflection points

Quoting from wikipedia inflection point, If the second derivative, $f''(x)$ exists at $x_0$, and $x_0$ is an inflection point for $f$, then $f''(x_0) = 0$, but this condition is not sufficient for ...
Heidegger's user avatar
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counterexample for a DC critical point that is not a limiting-stationary point?

Let $f$ be a DC function defined by $f = g - h$ where $g$ and $h$ are proper, lower semicontinuous and convex functions from $\mathbb{R}^n$ to $\mathbb{R}\cup\{+\infty\}$. A point $x^*$ is called a DC ...
kaienfr's user avatar
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1 answer
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Stationary points of a cubic function

If t is a positive constant, find the local maximum and minimum values of the function $f(x) = (3x^2 - 4)\left(x - t + \frac{1}{t}\right)$ and show that the difference between them is $\frac{4}{9}(t + ...
Gill Dixon's user avatar
1 vote
1 answer
194 views

Finding stationary points

Find the stationary points of $f$ of the minimization problem $$min_{x\in\mathbb{R}^2} 100(x_2 − x_1^2)^2 + (1 − x_1)^2$$ and determine which points are local and which global extremas. Problem: I ...
Uhmm's user avatar
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2 answers
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Assume the function has a maximum. Find it. $f(x,y,z)=3-x^2-2y^2-3z^2-2xy-2xz$ [closed]

Putting the first derivatives equal to 0 I get a critical point at C(0,0,0), and then using the second derivatives I get a Determinant of the Hessian matrix=-24. Is C the maximum? I have a doubt about ...
Jason's user avatar
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Stationary points of $2-\cos(\sqrt{x^2+y^2})$

I'm a bit confused with this. The gradient of the function is $$\nabla f=\left(\frac{x\sin A}{A} \quad \frac{y\sin A}{A}\right)^T $$ where $A=\sqrt{x^2+y^2}.$ One seemingly obvious solution of $$\...
Kündücs Eszkábál's user avatar
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Can we put bounds on the number of stationary points over parameter space for a vanilla MLP where loss is MSE?

Suppose $f(\vec{x}; \vec{\theta}, A)$ is a fully-connected vanilla (i.e. alternating matrix multiplications and sigmoid functions) multilayer perceptron that takes a vector $\vec x$ from the features ...
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2 votes
1 answer
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Making intuition about the Hamilton principle in classic mechanics?

I am trying to develop intuition about Why happen to be true the Hamilton's principle of stationary action, and after seen this video I have a few questions that are more related to maths than physics....
Joako's user avatar
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1 answer
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Calculate local minima and maxima

Consider the function $f : \mathbb{R^2} → \mathbb{R}$ defined by $$f(x_1, x_2)=e^{-(2x_1^2+3x_2^2)}$$ Determine whether the stationary point is a strict local maximum or a strict local minimum. $$$$ ...
Techlover's user avatar
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1 answer
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Calculate stationary points and local maxima and minima

Let $a, b∈ \mathbb{Z}$ \ {$0$} and let $f: \mathbb{R^2}→\mathbb{R}$ be defined by $$f(x_1, x_2)=ax_1^2+bx_2^2-4ab^2x_1-2a^2bx_2.$$ Find all stationary points of $f$ and, if possible, determine which ...
Techlover's user avatar
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Proof for stability of stationary point

Given a twice continuously differentiable function $f$ used for a difference equation $x_{n+1} = f(x_n)$, we can show that a stationary point $f(s) = s$ is asymptotically stable (see e.g. here for ...
NightRain23's user avatar
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830 views

What is the condition for the saddle point of a function of three variables?

For a function $f(x,y)$ of two real variables $x$ and $y$, a point $(x_0,y_0)$ is a saddle point if the determinant of the Hessian matrix $$[f_{xx}f{yy}-(f_{xy})^2]_{x=x_0,y=y_0}<0.$$ If we are ...
Solidification's user avatar
2 votes
1 answer
73 views

What is the meaning of "local optimality" in NLP mentioned in Constraint Qualification?

Most statements of Constraint Qualification I have found in the literature mention a locally "locally optimal solution" of the problem: $$ \begin{cases} \min f(x) \\ \text{s.t.}\\ g_i(x)\leq ...
shnnnms's user avatar
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1 vote
2 answers
114 views

Stationary and saddle points for $f(x,y) = x^3+x^2-xy+y^2+5$

Find Stationary and saddle points for $f(x,y) = x^3+x^2-xy+y^2+5$ What I have tried: $$\begin{align}f_x &= 3x^2+2x-y \\f_y &= -x+2y \\ \implies x&=2y,y = x/2 \end{align}$$ Plugging in the ...
dollar bill's user avatar
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0 answers
64 views

confusion about the meaning of a stationary integral

In calculus if variation, There are problems where I have to find the function (curve) that makes the value of an integral minimum between two points. It has to be stationary, in a sense that an ...
Ahmed Samir's user avatar
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1 answer
902 views

Calculating Stationary Points of $f(x,y)=e^{-(x^2+y^2)}$

Here's my attempt so far: $$f_x(x,y)=-2xe^{-(x^2+y^2)}$$ $$f_y(x,y)=-2ye^{-(x^2+y^2)}$$ I tried equating both the partial derivatives to $0$, and the only solution I seem to get is $(x,y)=(0,0)$. Are ...
Scripto Meow's user avatar
1 vote
1 answer
134 views

Finding extreme values of $ f :\mathbb R^2\rightarrow \mathbb R$, when the determinant $\Delta = AC - B^2 = 0$.

We generally rely on the result: [Let $ f$ be a real valued function from $\mathbb R^2$ with continuous partial derivatives at a stationary point $\vec a$ in $\mathbb R^2$. Let $A = D_{11}f(\vec a)$, $...
Benjamin Kurian's user avatar
4 votes
0 answers
227 views

The direction of the steepest descent path at the saddle point (Picard-Lefschetz theory)

I am having to perform oscillatory integrations like $e^{iS}$ using Picard-Lefschetz theory. One can write this as $e^{h+is}$ where $h(x,y)=-{\rm Im}(S(x,y))$ is the Morse function. To perform these ...
Faber Bosch's user avatar
1 vote
0 answers
56 views

Show $u(x)=\tan{(x/\sqrt{2})}$ is a solution to $0= u_{xx} + \frac{1}{2}u(1-u^2)$

I come across this problem in my study for diffusion-action nonlinear PDE. I tried to solve the problem explicitly but I stuck as I am going to show. However I solved the solution 2. I am not ...
Mr. Proof's user avatar
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1 vote
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Stationary points of a function defined on a manifold

I'm searching for stationary points of a multi-variable function which is defined on a manifold. To do this I parametrize my manifold and then differentiate the function with respect to the associated ...
AntoineM's user avatar
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333 views

Find the stationary points of the function $f(x) = x^2 + y^2$ subjected to constraints

Find the stationary points of the function $f(x) = x^2 + y^2$ subjected to the constraint $$ x^2 + y^2 + 2x - 2y +1 =0.$$
daniel marvin's user avatar
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0 answers
3k views

Is ${\bf F}_0={\bf V}_0 {\bf V}_0^H$ a locally optimal solution of $f(\bf F)$ if ${\bf V}_0$ is a locally optimal solution of $f({\bf V} {\bf V}^H)$?

${\bf F} \in {\mathbb C}^{N \times N}$ is positive semidefinite matrix and satisfies ${\text {tr}} ({\bf F}) \leq P$. $f({\bf F}): {\mathbb C}^{N \times N} \rightarrow {\mathbb R}$ is a real-valued ...
Hao's user avatar
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1 vote
1 answer
236 views

Show that $f(x,y)=x^2+4y^2-4xy+2$ has an infinite amount of stationary points

$f(x,y)=x^2+4y^2-4xy+2$ So, $f_x=2x-4y$ and $f_y=8y-4x$ To find the stationary points we have to equal the partial derivatives to $0$: $2x-4y=0$ $8y-4x=0$ Because we cannot find an $x$ and $y$ via the ...
EL02's user avatar
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2 votes
2 answers
187 views

Find the maximum and minimum of a multivariable function on a circle

This question is a continuation from a previous question I recently asked: Stationary points of a multivariable function I now have to find the maximum and minimum values of my function on the circle: ...
Charlie P's user avatar
  • 253
1 vote
2 answers
90 views

Stationary points of a multivariable function

This question might just be a quick one but I'm slightly confused by the answer I've been provided for this question. I have the function: $f(x,y) = (x^2+2y^2)e^{-y^2 - x^2}$ I found the partial ...
Charlie P's user avatar
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2 votes
0 answers
74 views

Long time decay of wave equation : why is it $t^{-(n-1)/2}$ instead of $t^{-n/2}$?

Consider the wave equation $$\begin{cases} \partial_t^2 u &= \Delta u, t \in \mathbb R \\ u(0) &= u_0 \in S'(\mathbb R^n) \\ \partial_t u(0) &= u_1 \in S'(\mathbb R^n) \end{cases}$$ where $...
Desura's user avatar
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1 answer
179 views

How do I find g(x) given the stationary points (2,9)?

question image given the info I form the equation g(x) = AX³+BX²+AX+C and did g'(x) = 0 to find B = -13A/4 (when x = 2). So using B and putting it into g(x) = 9, I get AX³-13A/4X²+AX+C = 9 and I am ...
GUTS mc GORDAN's user avatar
1 vote
1 answer
334 views

How to find stationary points of a multivariate quadratic function?

I'm given a function $$f = 391 x^{2} + 156 x y - 222 x z + 144 x w + 1224 x + 524 y^{2} - 156 y z - 88 y w - 2568 y + 391 z^{2} - 144 z w + 1016 z + 374 w^{2} - 1692 w$$ I find its stationary points ...
student's user avatar
  • 422
1 vote
1 answer
115 views

What is the problem in taking the derivative of a function with respect to a non-monotonic function?

I was going through the accepted solution here Danger Zone for Aircraft. And this is what caught my attention: So let's continue with the reasoning. We need to find the furthest possible distance x ...
Anonymous's user avatar
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1 vote
1 answer
98 views

How to efficiently find the max of this function?

I am a software developer and my software uses a function that I believe is very inefficient. I need to find the value of x that results in the maximum value out of the function. Currently, I will ...
Undead8's user avatar
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0 answers
163 views

Stationary phase method for two stationary points

I have a function $\psi(t)$ which has two stationary points at $t=a$ and $t=b$ and I want to find the asymptotic form of the integral $I(x)=\int_{a}^{b}f(t)e^{ix\psi(t)}dt$. Bender&Orzsag talks ...
123infinity's user avatar
0 votes
1 answer
65 views

Classifying the stationary point of $h(x,y,z) = 2(x−1)^2 + 3(y−1)^3 + 4(z−1)^4$

Given the function $h(x,y,z) = 2(x−1)^2 + 3(y−1)^3 + 4(z−1)^4$, I have found the only stationary point to be $(1,1,1)$. I then attempted to use the Hessian matrix to find out whether $(1,1,1)$ is a ...
toffeering's user avatar