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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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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 ...
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14 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, ...
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190 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 ...
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19 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 ...
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2answers
55 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} ...
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20 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 ...
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How to classify the degenerate critical (stationary) points of a multivariate function?

I have a multivariate function in one of whose critical points the Hessian matrix is singular. Is there any general method to determine the type of this critical point? Would be worth plotting the ...
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3answers
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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
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38 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 ...
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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 ...
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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 ...
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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 $f'(x) = 0$.
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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 $\frac{∂f}{∂y}$ = $...
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4answers
67 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 $P'(6)$. ...
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1answer
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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 ...
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1answer
59 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 ...
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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 $...
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35 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\...
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46 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 $ \...
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1answer
26 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 ...
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35 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 ...
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1answer
153 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}\...
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1answer
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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
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1answer
190 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 ...
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45 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 ...
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51 views

A stationary phase like lemma

Let $n \geqslant 3$, and $\phi : \mathbb{R}^n \to \mathbb{R}$ be a smooth function, and assume that $\phi \in L^1$ for $s > n/2$. I would like to prove the following estimate : For all $\lambda ...
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1answer
50 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)‎‎...
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1answer
293 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 ...
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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 ...
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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 ...
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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 \...
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113 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 ...
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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 ...
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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+...
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1answer
38 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 ...
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1answer
83 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 ...
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140 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. ...
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160 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) \...
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2answers
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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 ...
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81 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 $K = \{(x,y)\in\mathbb{R}^2 : 0\le x \le \pi , 0\le y \le 2\pi \}$...
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1answer
54 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 ...
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70 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_{...
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171 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 ...
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1answer
50 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 ...
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Debate on definition of Critical point and local / absolute extremas

So my question is: Is a global extremum necessarily a also local extremum? I think the answer is no if you define that f has a local max at c if f(c) ≥ f(x) for all x near c, even for those x not ...
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1answer
220 views

Why is $ dx/dy$ not a maths error if $dy/dx $ is $0$?

If $y$ is not changing with respect to $x$, then we will have a line $y=something$. At the same time, although $x$ occupies all the $x$ values possible, would it be correct to say that $x$ is not ...
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3answers
321 views

This second derivative is showing a point of inflection rather than a minimum point

For the curve $y=(2x-1)^4$, the derivative $8(2x-1)^3$ shows that the only stationary point is $(0.5, 0)$, which means it just touches the $x$-axis at that point. However, when determining the nature ...
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1answer
37 views

What is the name of the theorem/method for classifying extremum using second derivatives?

Given a formula $f : \mathbb{R} \rightarrow \mathbb{R}$ we know that by the Interior Exrtremum Theorem that, if $f$ attains an extremum at $c \in \mathbb{R}$, then $f'(c) = 0$. We also know that if $...
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73 views

Given critical points, find the original function of two variables

I am unsure how to attempt this question as it entails two variables. Find a function $f:ℝ^2→ℝ$ that has local extrema at $f(x,y)$ for all $[x,y]∈P$ $$P={\{[-8,5],[-7,6],[-5,2],[-3,-4],[5,5]}\}$$ ...
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1answer
2k views

Classification of critical points for two variables — determinant vs definiteness

In first year we were taught to classify stationary points using the determinant of the Hessian matrix -- which was procedural and simple enough. In second year we were introduced to classifying ...