Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

0
votes
0answers
8 views

Stationary phase approximation higher order

I want to evaluate the following integral for large $z:$ \begin{eqnarray} I(r,z)=\int_{0}^{\infty}d\rho e^{i\sqrt{1-\rho^2}z} J_0(r\rho)J_1(R\rho) \label{eq:fourier:integral} \end{eqnarray} I ...
2
votes
1answer
33 views

Is there a functional (i.e. infinite-dimensional) generalization of the second partial derivative test?

For a smooth function $f: \mathbb{R}^n \to \mathbb{R}$, we can (usually) test whether a critical point ${\bf x}_0$ (at which ${\bf \nabla} f({\bf x}_0) = {\bf 0}$) is a local maximum, minimum, or ...
-2
votes
0answers
39 views

Function approximating camels humps?

I want to construct a function with two maxima and a minima in between that would approximate the two humps of a camel. In addition, I would like that there would be some effort (not a polynomial) to ...
0
votes
1answer
28 views

Second Partial Derivative Test using Hessian Determinant

I understand that the Hessian determinant (detH) for a function f (x,y) is defined as: \begin{vmatrix} f_{xx} & f_{xy}\\ f_{yx} & f_{yy}\\\end{vmatrix} Where the determinant is a factor for ...
0
votes
1answer
13 views

Finding the stationary point of a type of hyperbola?

I know that to find stationary points on a function, we need to differentiate the function and set that = 0. But how can we find the stationary points of the below function? $$y=\frac{1}{x} + \frac{...
0
votes
1answer
26 views

Find and classify the stationary points of $y = x^ 2/(x-4)$

I have already obtained the answer by using the quotient rule and so there are stationary points at x = 0 and x = 8. I am fine at doing these types of questions but I don't fully understand the ...
0
votes
2answers
69 views

Find the coordinates and nature of the stationary point of $z=x^3+y^3-6xy$

Find the coordinates and nature of the stationary point of $$z=x^3+y^3-6xy$$ So I have found all the partial derivatives but I'm not sure how to then find the stationary point. All I know is $\frac{\...
1
vote
0answers
31 views

Stationary phase for retarded potentials in electromagnetism

I want to apply something like a stationary phase approximation to the following expression $\int_V d^3x' \frac{B(x')}{|x-x'|}e^{ik|x-x'|}$ with $x\in \mathbb{R}^3$, $k\rightarrow \infty$ and $B$ is ...
1
vote
1answer
20 views

Find the stationary points of: $V=(2/r^3)-(3/r^2)$

I've applied the exponent rules and differentiated to get to the point where i have $$dr/dv=-6r^{-4}+6r^{-3}$$ I'm starting to get very confused with the math when i set the LHS = 0 in order to ...
0
votes
0answers
17 views

Queuing theory for may task in univ

A gas station only has one pump for refueling the Pertamax type. The arrival of Pertamax-fueled cars to the gas station follows the process Poisson with an arrival rate of 15 cars / hour. However, ...
7
votes
3answers
193 views

Inflection point for function with fractional exponents

Show that $f(x) = 4x^{1/3}-x^{4/3}$ has an inflection point at $x=1$. I correctly get $$f'(x) = \frac{4(1-x)}{3x^{2/3}}\implies f''(x)=-\frac{4(x+2)}{9x^{5/3}}$$ It is clear to me that there is an ...
0
votes
0answers
21 views

Directional derivatives at the stationary points

For z=f(x,y) , in the stationary points, the partial derivatives with respect to x and y are both 0. But are all the directional derivatives of the stationary points zero as well? Why do we only care ...
0
votes
2answers
73 views

Classifying stationary points in 3 variable case

I have a following problem and I am not sure if I understand correctly how to classify stationary points. The function is given by: \begin{equation} f(a, b, c) = a^2b + b^2c + c^2a, \end{equation} ...
0
votes
1answer
22 views

Find and classify the stationary points for $f(x,y) = (4x_1^2 - x_2)^2$

So first I calculated the gradient which was $\nabla f(x) = (64x_1^3 - 16x_1x_2, -8x_1^2 + 2x_2)^T.$ Then setting the equations in this equal to $0$ I solved for $x_2$ and got $x_2=4x_1^2$. Then ...
1
vote
3answers
59 views

Find the values of $m$ such that the polynomial $P(x) = x^3-mx^2+12x+11$ has no turning points

$P(x) = x^3-mx^2+12x+11$ differentiate: $y' = 3x^2 -2mx +12$ After this point I'm unsure how to progress
0
votes
2answers
41 views

Why doesn't conical surface have a stationary (critical) point (at 0,0)?

Function:$$x = {- \sqrt{x^2+y^2}}.$$(a conical surface) To determine whether it has a stationary point or not, 2 condition must be met: function must have partial derivatives at point p0, and ...
1
vote
0answers
31 views

Is there a parallel between extremal solutions to Lagrangians and stationary points of a real function?

Background In reading about (a simple version of) Noether's theorem on invariances, I've been told that we should seek stationary points of the Lagrangian functional $\int_a^b L(\gamma)\,dt$, and ...
1
vote
2answers
50 views

Need help with a calculus maximum question

The question is: Which values of $k$ give a maximum at $x=−1$ for $f(x)=(k+1)x^4−(3k+2)x^2−2kx$? I found that $f'(x)=4x^3k+4x^3-6xk-4x-2k$, but I'm confused on where to go from here because ...
-2
votes
2answers
39 views

Showing point in f(x) such that f'(x)=0 exists [closed]

let $f(x)=x^4 + \sin x$ Show that there exists $x \in (-2,2)$ such that $f'(x) = 0$.
0
votes
2answers
38 views

Location and nature of all the stationary points of function

Just wanted to check if this was right before I proceed f(x,y)=$2x^3 + 6xy^2 - 3y^3 - 150x$ which gives $\frac{∂f}{∂x}$ = $6x^2 + 6y^2 -150$ Then doing the same with y gives $\frac{∂f}{∂y}$ = $...
2
votes
4answers
68 views

Why is $2$ double root of the derivative?

A polynomial function $P(x)$ with degree $5$ increases in the interval $(-\infty, 1)$ and $(3, \infty)$ and decreases in the interval $(1,3)$. Given that $P'(2)= 0$ and $P(0) = 4$, find $P'(6)$. ...
0
votes
1answer
19 views

Proving that a function in $\mathbb{R^3}$ has a minimum on a set

Given $$f(x,y,z)=2x^2+y^2+z^2$$ I have to prove that f has a minimum on the set $$E=\{(x,y,z)\in \mathbb{R^3}: x^2yz=1\}.$$ $E$ is a closed set because of the fact that $g(x,y,z)=x^2yz$ is the ...
0
votes
1answer
61 views

What does equation of error function to zero exactly do?

I've seen this note and got more improved understanding of how relative extrema could be found in calculus. Professor Strang in his linear algebra course explains how function could be minimized by ...
1
vote
0answers
94 views

Bifurcation, critical points and parameter dependence in $\nabla_r\,\phi_a(r) = 0$ when varying $a$

Suppose I have a function $\phi_a(r)$, where $r \in \mathbb{R}^n$ denotes a real n-dim. vector, and where $a \in \mathbb{R}^p$ denotes a set of additional real parameters. Suppose that for a given $...
0
votes
0answers
39 views

Is there a simple way to demonstrate $y'(x)<0$ where $y = \frac{(1-x)(2-x)x - (2-x)^2 \log(2-x)}{x(x-4)(x-1)^2}$

If we define $y(x)$ such that: $$y = \frac{(1-x)(2-x)x - (2-x)^2 \log(2-x)}{x(x-4)(x-1)^2},$$ is it possible to demonstrate that $y'<0$ for $x\in(0,1)$? You can easily show that $\lim_{x\...
0
votes
0answers
64 views

saddle point optimality conditions

the problem $\min \{ \nabla f(x_k)^t\delta+\frac {1}{2} \delta^tH(x_k)\delta \}$ where $f : \mathbb R^n \to \mathbb R$ and $ H(x_k)$ is hessian of f at point $x_k$ and suppose that $ \...
1
vote
1answer
29 views

Finding stationary points of a function when substitution method fails

I have a function, $f(x,y)=x^2+2xy+y^2+\alpha x+\beta y$ where $\beta$ and $\alpha$ are parameters. I'm asked to find the stationary points of the function so naturally, I tried to find where the ...
3
votes
1answer
40 views

Number of equilibrium points, first order ODE

Question: Determine the number and location of the equilibrium points for all values of $c$ of $\dot{x}=x^2+2x+c+2$, $(1)$ $x \in \mathbb{R}$, where $c \in \mathbb{R}$, constant, is a control ...
4
votes
1answer
164 views

Convergence concerning the $\alpha$th derivative of $f(x)=x^\alpha-\alpha^x$

Let $x_0$ be the stationary point of the $\alpha$th derivative of the function $f(x)=x^\alpha-\alpha^x$, and let $$\lambda_\alpha=\frac{f(-x_0)}{-x_0}.$$ Does the limit $$\lim_{\alpha\to\infty}\...
-1
votes
1answer
21 views

Why is a stationary point of a curve given by parametric equations on a surface not a stationary point of the function itself?

I am fine with most of this question but am unsure about the stationary point not being a stationary point of the function? The question
1
vote
1answer
262 views

Finding the largest area of a right-angled triangle using Lagrange multipliers

The area of a triangle of length a, b and c is given by $$A =\sqrt{s(s-a)(s-b)(s-c)}$$ where its perimeter is $2s$ such that $2s = a + b + c$ Consider a right-angled triangle with hypotenuse $a$ such ...
0
votes
0answers
55 views

Stationary Condition of Variational Iteration Method

Would you kind help me? I am an undergraduate student and i am studying about variational iteration method (VIM) by Ji Huan He. But, when i study about the stationary condition of VIM, there is a ...
2
votes
1answer
51 views

Equilibrium of autonomous first-order differential equation: ‎

Let ‎$ ‎‎(\bar{x},‎\bar{y},‎\bar{z})‎‎‎‎ $‎ be an equilibrium of autonomous first-order differential equation: ‎\begin{equation}‎ ‎\begin{cases}‎ ‎\dot{x}=‎ f_1(x,y,z)‎‎‎,\\‎ ‎\dot{y}=‎ ‎f_2(x,y,z)‎‎...
6
votes
1answer
333 views

An unexpected application of the Cauchy-Schwarz inequality for integrals

I discovered this yesterday and I just want to know whether my solution is correct and whether there's a shortcut to it. Let $g(x)$ be a twice-differentiable continuous function that crosses the ...
2
votes
0answers
60 views

Conditions for $a,b \in \mathbb R$, such that $0 \in \mathbb R^2$ is exponentially stable for the system : $x_1^+=x_2, x_2^+=ax_1 + bx_2$

Exercise : Find competent and necessary conditions for $a,b \in \mathbb R$, such that $0 \in \mathbb R^2$ is an exponentially stable stationary point for the the linear dynamical system of discrete ...
0
votes
3answers
39 views

Does a stationary point of $f(x, y)$ need to have a zero slope in every direction?

Does a stationary point of $f(x, y)$ need to have a zero slope in every direction? If so, then why is it sufficient to show $ $ $ $ $f_x(a, b)=f_y(a, b)=0$ $ $ $ $ to determine that $(a, b)$ is a ...
10
votes
3answers
1k views

Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

I've just come across this function when playing with the Desmos graphing calculator and it seems that it has turning points for many values of $a$. So I pose the following problem: Given $a \in \...
0
votes
3answers
128 views

Do inflection points of $f(x)$ give $f'(x)=0$?

I understand that the values of $x$ that allow $f'(x)=0$ are stationary points and therefore potential local maximums and minimums of $f(x)$. When would a stationary point NOT be a local maximum or ...
1
vote
2answers
46 views

Instance where echelon form of a matrix conceals the solution to a system?

Say I have two equations: $$y^3+x-y=0$$ $$x^3-x+y=0$$ These are both partial derivatives of some function, and the condition that they're equal to $0$ implies the values of $x,y,z$ are bound are ...
0
votes
1answer
94 views

Stationary Points of : $x' = 7x + 10y + 3, y' = -5x -7y + 1$

Exercise : Find the Stationary Points and then their kind and behavior, of the system : $$x' = 7x + 10y + 3, y' = -5x -7y + 1$$ Attempt : By solving the linear system : $$\begin{cases} 7x+...
0
votes
1answer
40 views

Number of stationary areas for polynomial of many variables

Suppose we have a polynomial $f$ of $k$ variables of degree $n$. $\mathcal S$ is a set of stationary points, on which equation $\nabla f=0$ holds. What is the maximum number of connected components of ...
0
votes
1answer
84 views

Stationary points of : $y' = y^3 + y$

I have a question regarding stationary points and their behavior. Exercise : Discuss the behavior of the stationary points of the differential equation : $$y' = y^3 + y$$ Attempt : We can ...
0
votes
1answer
186 views

what are the stationary points for the following 2 variable function

f(x,y) = (y^2 -3y) sinx in the open rectangle 0 < x < 2pi and 1 < y < 5. i have found (pi, 3), (pi/2, 3/2) and (3pi/2, 3/2) which are saddle point, local min and local max respectively. ...
0
votes
2answers
227 views

Check if function is coercive

I am trying to check if the function $U(x,y)=6\ln x+\ln y$ is coercive. I know, that i need to check if $\lim_{\vert (x,y) \vert \rightarrow \infty} U(x,y) = +\infty$, and so far I have $\lim_{(x,y) \...
1
vote
2answers
62 views

Proving that a function has exactly four stationary points

I have the following function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ defined by $$f(x,y)=xy(x+y-1).$$ Using the property that the gradient in the stationary points is zero I calculated the four ...
0
votes
3answers
100 views

Critical point of a function with $\sin$ and $\cos$

I have to find and classify all the critical points of the function $f: K\to \mathbb{R}$ $$f(x,y) = \sin(2x)\cos(y)$$ inside the set $K = \{(x,y)\in\mathbb{R}^2 : 0\le x \le \pi , 0\le y \le 2\pi \}$...
0
votes
1answer
62 views

Fermat stationary points proof using definition of derivatives: don't get the limit claim

I don't understand in the proof below why you can claim that the derivative the function in $x_0$ of $h\to 0$ for $h > 0$, is smaller than or equal to zero. From what rule this follows? Does it ...
0
votes
0answers
79 views

Nonlinear Matrix Equation Fixed Point Convergence

I have a method that searches for the fixed point of the equation $$x = A(x)x$$ where x is a vector and A(x) is a square matrix. I am able to convert this into a fixed point method $$x_{n+1} = A(x_{...
0
votes
3answers
205 views

Prove that the function has at least one stationary point in the set $(-1,1)$

Given a function $f:[-1,1]{\rightarrow\mathbb{R}}$ such that $f$ is continuous on $[-1,1]$ and differentiable on $(-1,1)$ and such that$$f(-1)=0, \space f(0)=-1, \space f(1)=2.$$ Prove that the ...
0
votes
1answer
57 views

Gradient vector orthogonal to set tangents

Let $A \subset \mathbb{R}^n$, with $A$ open, and let $\emptyset\ne S \subset A$. Let $f\in C^1 (A, \mathbb{R} )$ be a function. Let $\vec{a}$ be a point in $S$, and suppose that $\vec{a}$ is a local ...