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).
171 questions
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Is there a general formula for the stationary points of this multivariate polynomial?
Suppose we have a multivariate polynomial given by
$$
p(x,y) =
\begin{bmatrix}
1 & x & x^{2} & x^{3}
\end{bmatrix}
\begin{bmatrix}
a_{00} & a_{01} & a_{02} & a_{03} \\
a_{10} &...
- 1,201
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How do I find and classify a stationary point of the function $5x^3 - 3yx - 6y^3 - 2$ using Newton's method and the Hessian eigenvalues?
Setup to the problem:
We are going to determine the stationary points of the function
$5x^3 - 3yx - 6y^3 - 2$
in the region $-1 \leq x \leq 1, \ -1 \leq y \leq 1$.
Calculate the gradient $\nabla f(\...
- 407
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Stationary points - Gateaux vs Frechet derivative
In basic vector calculus one terms a point $f$ stationary for $E$ if $\nabla E(f) = 0$. On the other hand, in variational calculus we term $f$ stationary for $E$ if the first variations are zero at $f$...
- 2,408
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Finding the value of $x_0$ for $f(x)=e^{\sin(x)} + \sin(x)$ where $f(x_0)=0$ [closed]
This question is from a high school calculus course.
We had to find all critical points on an interval $[0,x_0]$ such that $f(x_0)=0$
For the function $$f(x)=e^{\sin(x)} + \sin(x)$$
Differentiating $f$...
- 69
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Relationships between $\frac{\partial^2f}{{\partial}x^2}$ and $\frac{\partial^2f}{{\partial}y^2}$ due to the nature of $|\mathbb{H}|_f$
$\text{ consider a continuous function of }x\text{ and }y\text{ ; }\;\;f:X\to{\mathbb{R}}^2, \;X{\subseteq}{\mathbb{R}}^2$
$\text{ with second partial derivatives that exist and are continuous,}$
$\...
- 390
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Stationary Solution to Hill's Equation of Motion
In Hill's problem for the three-body system, the dimensionless equations of motion can be written as follows:
$$
\frac{d^2\xi}{d\tau^{2}} - 2\frac{d\eta}{dt} = - \frac{\xi}{\rho^{3}} + 3\xi \, ,
\\
\...
- 465
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The set of values of $a$ for which the function $f(x)=(a^2-3a+2)\cos\left(\frac{x}{2}\right)+(a-1)x$ possess a critical point
The set of values of $'a'$ for which the function $f(x)=(a^2-3a+2)\cos\left(\frac{x}{2}\right)+(a-1)x$ possess a critical points
The set of values of $'b'$ for which the function $f(x)=(b^2-3b+2)\...
- 4,708
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Formalizing the "maximum weight" problem
tl;rd How to generalize the fact that the shape of Earth which maximizes your weight is one in which all points on the boundary of Earth contribute the same?
My favorite physics question (credit here) ...
- 135
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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 $x\to\infty$ ...
- 325
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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{...
- 395
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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 ...
- 3
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68
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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 $(...
- 103
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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{...
- 77
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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 ?
...
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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 ...
- 21
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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 ...
- 5,565
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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 ...
- 881
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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 ...
- 301
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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$$ and determine which points are local and which global extremas.
Problem: I ...
- 327
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95
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Stationary points of $2-\cos(\sqrt{x^2+y^2})$
I'm a bit confused with this.
The gradient of the function is $$\nabla f=\left(\frac{x\sin A}{A} \quad \frac{y\sin A}{A}\right)^T $$ where $A=\sqrt{x^2+y^2}.$
One seemingly obvious solution of $$\...
- 189
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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....
- 2,167
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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 ...
- 295
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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 ...
- 723
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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 ...
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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 ...
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