Mathematics Stack Exchange
Mathematics 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).

A stationary point is a point on a graph of a function where the derivative of the function is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). Stationary points are used for finding minimum and maximum values of functions within a domain and for curve sketching. This tag is for questions involving the existence and classification of stationary points. For questions focusing on minimizing of maximizing values under constraints, use .

171 questions
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Interpretation of eigenvectors of Hessian in context of local min/max/saddle?

Say $f \in C^2$ so we can possibly use its Hessian $H$ to determine whether $f$ has a local max, min, or saddle at a critical point $x_0$. Since $H(x_0)$ is real and symmetric, it is diagonalizable, say with eigenvector-eigenvalue pairs…
nullUser
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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 $a \in \mathbb{R}-\{0\}$, find $x$ such that…
TheSimpliFire
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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 inflection point at $x=-2$ since this value of…
user163862
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Convergence concerning the $\alpha$th derivative of $f(x)=x^\alpha-\alpha^x$

Let $x_\alpha$ be the stationary point of the $\alpha$th derivative of the function $f(x)=x^\alpha-\alpha^x$, and let $$\lambda_\alpha=\frac{f(-x_\alpha)}{-x_\alpha}.$$ For $\alpha\in\Bbb N$, does the limit…
TheSimpliFire
  • 28,020
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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 points $(0, t)$ and $(1, 1)$ and has stationary…
TheSimpliFire
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The direction of the steepest descent path at the saddle point (Picard-Lefschetz theory)

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 kinds of integrals one deforms the integration…
Faber Bosch
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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 two direction of descent: (0,-1) and (0,1).…
worldsmithhelper
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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 saddle point via the second partial derivative test,…
tparker
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Are all points $x$ with $f'(x) = 0$ local maxima, local minima, or stationary inflection points? + Possible Counterexample

In this post, I am considering only functions of the form $f:\mathbb{R}\rightarrow\mathbb{R}$. It's common to classify stationary points into local maximum points, local minimum points, and saddle points / stationary inflection points. Is this…
MaximusIdeal
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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{equation} \int_{-\infty}^{\infty}…
Leonardo
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Asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$?

What are asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$ ? Here is the graph for $n=10$. The turning point in $x\in(0,1)$ on…
Dan
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Question on estimate in one of Jean Bourgain's 1992 papers

The goal of the argument is to estimate the following integral: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\varphi_k(|\xi|)^2\,d\xi$$ where $t(x):x\in B(0,1)\mapsto t(x)\in(0,1)$ is arbitrary and $\varphi_k(|\xi|)$ is a…
Diffusion
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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 $k$ times differentiable with $k \mod 2 \equiv…
Ethan Mark
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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, $f(x)$, $f'(c) = f''(c) = 0$ and $f'''(c) \neq 0$, then $c$ is a…
Ethan Mark
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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 differentiating the equation to get $$…
AH_01
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