# 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).

128 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
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 $$x^2 + y^2 + 2x - 2y +1 =0.$$
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 ...
8 views

### Fractional stationary curve.

Define fractional stationary. Is there any relation between stationary curve and fractional stationary curve? please provide sufficient examples.
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 ...
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 ...
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: ...
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 ...
36 views

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 ...
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 ...
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 $x$-...
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 ...
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 ...
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 ...
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 ...
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 $\mathbb{R}^3$ ...
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 ...
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 ...
265 views

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 ...
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 ...
119 views

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, $f(x)$, $f'(c) =... 1answer 40 views ### How to prove that a function has a point of inflexion when the function is in terms of constants only? Below is the question that I have been working with: And here is the solution to part c), the part that I am stuck on: Here’s the thing, I understand why the first two factors are greater than zero (... 1answer 47 views ### Showing that$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 ... 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: $$\frac{dy}{dx}=\cos(2x)\... 1answer 113 views ### Is local minimum/maximum necessarily global when it's the only stationary point of a continuous & differentiable function? Couldn't find this theorem even though it feels very intuitive to me. If the f:R^n \to R is continuous, and has only one stationary point - a local minimum/maximuma. Doesn't it necessarily makes it ... 2answers 56 views ### Find the stationary points of f(x,y)=5y\sin(3x) Given the function f(x,y)=5y\sin(3x) find the stationary points. I found f_x=15y\cos(3x). Solving f_x=0, I got y=0,x=\frac{(2n+1)\pi}{6} Similarly, f_y=5\sin(3x). Solving f_y=0, I got x=... 1answer 80 views ### Steepest descent with multiple saddles I have an issue with application of steepest descent, especially in the presence of more than one stationary point, where it seems that deformation of the integration contour could take one through an ... 1answer 69 views ### Can a trajectory pass through a critical point for a plane autonomous system? (Differential equations) Set up: I am considering a plane autonomous system where there exists two ODEs, \frac{dx}{dt}=X(x,y),\frac{dy}{dt}=Y(x,y). We then usually draw trajectories on the phase plane to indicate the ... 1answer 179 views ### Infinite stationary points for multivariable functions like x*y^2 I have found several questions about functions with infinite stationary points like What if there are infinite stationary points? Find all stationary points of multivariable function Classifying ... 2answers 44 views ### For which natural numbers a the function f(x)=x^ae^x has exactly one extremum? I need to find for which natural values of a the function f(x)=x^ae^x has exactly one extremum. I calculated the derivative: f'(x)=e^xx^{a-1}(a+x), but I don't know what to do. I know that I can'... 0answers 46 views ### Multivariable calculus critical points I've got the following equation for all (x,y) in \mathbb{R}^2 where a is a real number:$$f(x,y) = 4ay^2-x^2y^3-x^2$$I want to calculate all critical points when:$a=0a>0a<0$If ... 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,...
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 ...