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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126 views

The direction of the steepest descent path at the saddle point

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 ...
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53 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 ...
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14 views

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 ...
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32 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 $$ x^2 + y^2 + 2x - 2y +1 =0.$$
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17 views

Why would the nonlinear term have different impacts on hyperbolic and non-hyperbolic stationary points?

Why would the nonlinear term ($o(|x|^2)$ in $\dot x = f(x) = Df(0) x + o(|x|^2)$) have different impacts on hyperbolic and non-hyperbolic stationary points? My guess is that for non-hyperbolic ...
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8 views

Fractional stationary curve.

Define fractional stationary. Is there any relation between stationary curve and fractional stationary curve? please provide sufficient examples.
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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 ...
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1answer
63 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 ...
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2answers
37 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: ...
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2answers
47 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 ...
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36 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 $...
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1answer
38 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 ...
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1answer
61 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 ...
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1answer
72 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 ...
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43 views

Find set of values such that stationary/inflection points exist.

Let f:→x∈ R I x^2+bx+1>0}→R defined by f(x)=loge(x^2+bx+1) where b∈R is a constant. a)Find the set S of values of b for which f has a stationary point. Show your reasoning. b)Find the set T of ...
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9 views

stationary vector for an unbreakable markov chain with period 3

I need to find an unbreakable markov chain with period 3 on all the natural numbers such that it's stationary vector $\Pi =(\pi_0,\pi_1,\ldots)$ follows: $\pi_1 = \pi_2 = 1/3$ my attempt was that $0$ ...
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17 views

Is it rigorous to judge the stationary point of a function by the nature of the function?

Consider the following function, $F(f_{1}, f_{2})$, where \begin{equation} F(f_{1}, f_{2})= p_{1}p_{2}\ln{[1+(u_{1} - 1)f_{1}+(u_{2} - 1)f_{2}]} \\+ (1-p_{1})p_{2}\ln{[1+(d_{1}-1)f_{1}+(u_{2} - 1)f_{2}...
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1answer
77 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 ...
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61 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 ...
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1answer
34 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 ...
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1answer
59 views

Classifying the stationary point of $f(x,y) = 4y^2 + 6yx^2 + 17$

Given the function $f(x,y) = 4y^2 +6yx^2 + 17$, I found the single stationary point $(0,0)$. I then tried to use the Hessian matrix to classify it as a local minimum/maximum/saddle point, but the ...
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1answer
28 views

Stationary point of $ x \mapsto \frac{1}{2} \| A x - b \|^2 + \frac{\lambda}{2} \| D x \|^2 $

Given full rank matrix $A \in \mathbb{R}^{n \times m}$, matrix $D \in \mathbb{R}^{p \times m}$, vector $b \in \mathbb{R}^n$ and scalar $\lambda > 0$, let scalar field $f: \mathbb{R}^n \to \mathbb{R}...
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25 views

Finding stationary point of vector-valued function

I'm trying to find a stationary point of the function $r(u) = \gamma(\begin{matrix} \alpha -u_1+u_1^2u_2\\ \beta - u_1^2u_2 \end{matrix})$ , with $ \alpha , \beta , \gamma > 0 $ I have taken ...
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18 views

Does this non-negative function without stationary points have only descend directions close to a constraint set?

Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0} $ is differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...
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1answer
76 views

Show $x$-coordinate of stationary point satisfy the equation $x=\frac32+ \frac{x}{e^{2x}}$

I solved the equation but when it satisfies it spouse to have opposite signs i think this is the question Question 1 The curve with equation $y=\dfrac{e^{2x}+x}{x^3}$ has a stationary point with $x$-...
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5answers
69 views

how do you find the $x$ value for $-\sin x+\cos x=0$

Find the sationary points of the curve and their nature for the equation $y=e^x\cos x$ for $0\le x\le\pi/2$. I derived it and got $e^x(-\sin x+\cos x)=0$. $e^x$ has no solution but I don't know how ...
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2answers
437 views

Find the x-coordinate of the stationary points of the curve and determine the nature of these stationary points.

The equation of a curve is $y=x^2e^{-x}$. Find the x-coordinate of the stationary points of the curve and determine the nature of these stationary points. Show that the equation of the normal to the ...
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19 views

Determine if $f(x,y)=(1+\sin(x+y)) \ln(1+2x+y)-2x-y$ has a maximum at the origin

I want to determine if the function $f(x,y)=(1+\sin(x+y)) \ln(1+2x+y)-2x-y$ has a local extrema at the origin, and if so determine its characteristic. I found the quadratic form of the function ...
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0answers
32 views

Characterizing (stationary) points by the number of valleys one can descent into

In non-convex optimizing of more than 2 times differentiable $f: \mathbb{R}^2 \mapsto \mathbb{R}$ we can encounter saddle points that have multiple valley one could descent into. At $(0,0)$ there are ...
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1answer
27 views

Maximum on circle through normal p.d.f in $\mathbb{R}^3$

Given a normal distribution on $\mathbb{R}^3$ $$p(\mathbf{x})\propto\exp(-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^T\Sigma^{-1}(\mathbf{x}-\mathbf{\mu}))$$ and a circular trajectory through $\mathbb{R}^3$ ...
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29 views

Are stationary points preserved when squaring a function under an integral?

Say $T(b)=\int_0^{10}f(b,x)dx$. I want to find the value of $b$ which minimises $T$ but evaluating that integral is quite difficult. If $f(b,x)$ was squared, however, it would be easier. So my ...
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1answer
80 views

$f$ have finitely many critical points in $\Omega$

Assume that $\Omega$ is an bounded open set in $R^m, f\in C^2(\overline{\Omega},R^m)$. If $f$ does not have any critical point in $\partial \Omega$, and all the critical points of $f$ in $\Omega$ are ...
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1answer
265 views

Find Stationary Point(s) for function (two variables): $f(x,y)=3y^3-x^3-2y^2+4x-2y$

Find all stationary pointsfor function $$f(x,y)=3y^3-x^3-2y^2+4x-2y.$$ So far this is what I have $$\frac{\partial f}{\partial x}\left(3y^3-x^3-2y^2+4x-2y\right)=-3x^2-4$$ and $$\frac{\partial f}{\...
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0answers
45 views

Cross-entropy loss and stationary points

I am trying to find the stationary points of the cross-entropy function for binary classification : $$ L(w) = -y \cdot \log(\sigma(wx)) - (1-y) \cdot \log (1-\sigma(wx)) $$ with $$ \sigma(wx) = \frac{...
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2answers
168 views

Constrained optimisation: stationary points of constrain

I'm new to optimisation and have a problem. I'm supposed to find stationary points to the following function $f$ under the constrain $g$: $$f(x,y) = xy$$ $$g(x,y) = x^4 + y^4 + 2xy - 4 = 0$$ which ...
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1answer
121 views

Proving stationary points of inflection

Edit For the purposes of proving the statement below, a stationary point of inflection of a curve shall be defined as a point on the curve where the curve changes concavity. Problem Suppose $f(x)$ is ...
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1answer
119 views

On locating inflection points

From what I have learnt, a point of inflection of a curve is, by definition, a point where the curve changes concavity. The Simple Case Thus, if, for a point, $c$, on a given function, $f(x)$, $f'(c) =...
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1answer
40 views

How to prove that a function has a point of inflexion when the function is in terms of constants only?

Below is the question that I have been working with: And here is the solution to part c), the part that I am stuck on: Here’s the thing, I understand why the first two factors are greater than zero (...
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1answer
47 views

Showing that $y = \frac{(x-a)e^x}{(x-b)}$ has stationary points when $a-b<0$ or $a-b>4$

I have the function $$y = \frac{(x-a)e^x}{(x-b)}$$ and I am told that the curve has stationary points under the following conditions - $$a-b < 0 \quad\text{or}\quad a-b>4$$ I started by ...
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1answer
196 views

Find the x-coordinate of the stationary point on the curve $\tan(x)\cos(2x)$ for $0 < x < \pi/2$

Can someone please show me how to find the x-coordinate for the stationary point for this curve? $y=\tan(x)\cos(2x)$ for $0 < x < \pi/2$ This is what I've done so far: $$\frac{dy}{dx}=\cos(2x)\...
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1answer
113 views

Is local minimum/maximum necessarily global when it's the only stationary point of a continuous & differentiable function?

Couldn't find this theorem even though it feels very intuitive to me. If the $f:R^n \to R$ is continuous, and has only one stationary point - a local minimum/maximuma. Doesn't it necessarily makes it ...
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2answers
56 views

Find the stationary points of $f(x,y)=5y\sin(3x)$

Given the function $f(x,y)=5y\sin(3x)$ find the stationary points. I found $f_x=15y\cos(3x)$. Solving $f_x=0$, I got $y=0,x=\frac{(2n+1)\pi}{6}$ Similarly, $f_y=5\sin(3x)$. Solving $f_y=0$, I got $x=...
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1answer
80 views

Steepest descent with multiple saddles

I have an issue with application of steepest descent, especially in the presence of more than one stationary point, where it seems that deformation of the integration contour could take one through an ...
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1answer
69 views

Can a trajectory pass through a critical point for a plane autonomous system? (Differential equations)

Set up: I am considering a plane autonomous system where there exists two ODEs, $\frac{dx}{dt}=X(x,y),\frac{dy}{dt}=Y(x,y)$. We then usually draw trajectories on the phase plane to indicate the ...
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1answer
179 views

Infinite stationary points for multivariable functions like x*y^2

I have found several questions about functions with infinite stationary points like What if there are infinite stationary points? Find all stationary points of multivariable function Classifying ...
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2answers
44 views

For which natural numbers $a$ the function $f(x)=x^ae^x$ has exactly one extremum?

I need to find for which natural values of $a$ the function $f(x)=x^ae^x$ has exactly one extremum. I calculated the derivative: $f'(x)=e^xx^{a-1}(a+x)$, but I don't know what to do. I know that I can'...
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0answers
46 views

Multivariable calculus critical points

I've got the following equation for all $(x,y)$ in $\mathbb{R}^2$ where $a$ is a real number: $$f(x,y) = 4ay^2-x^2y^3-x^2$$ I want to calculate all critical points when: $a=0$ $a>0$ $a<0$ If ...
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1answer
153 views

Proving $(0,0)$ is a saddle point for $f(x,y)=2y^3-6y^2+3x^2y$

The function $f(x,y)=2y^3-6y^2+3x^2y$ has 2 stationary points, $(0,0)$ and $(0,2)$. Using the function's Hessian I managed to prove that $(0,2)$ is a strict local minima, but the Hessian of $f$ at $(0,...
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2answers
88 views

Find the stationary points

Determine the stationary points of the following function and for each stationary point determine whether it is a local maximum, local minimum or a point of inflexion. $f(x)=x^3(x-1)^2$ Using ...
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1answer
24 views

Find the stationary point and its nature.

To find the stationary point I have to find dy/dx=0 so far I reached until this point where I'm not sure how to get the 2 X values: dy/dx= how to find the stationary points from here?