# 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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### First & second derivative of $g(t)=f(\alpha+t\cos\theta,\beta+t\sin\theta)$ and conditions for minimum

If $f(x,y)$ is a smooth real-valued function, and $g(t)=f(\alpha+t\cos\theta, \beta+t\sin\theta)$ express $g’(t)$ and $g’’(t)$ in terms of partial derivatives of $f$. Then deduce the conditions for $t$...
1 vote
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### Method of Steepest Descent (deform contours where there are 2 saddles)

Question: use the method of steepest descent to obtain the first two non-zero terms in the asymptotic approximation $$\int_0^\infty \exp(ix(t^3/3+t))dt\sim i(1/x+2/x^3+...+a_n/x^n)$$ as $x\to\infty$ ...
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### Classify stationary points of $f(x) = 2x_1^3 - 6x_2^2 + 3x_1 ^2x_2$

Given be the function $f(x) = 2x_1^3 - 6x_2^2 + 3x_1 ^2x_2$. I would like to compute and classify all stationary points $x^*$. \begin{align*} \text{Function:} \ f( x) & \begin{array}{l} =2x_{1}^{...
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1 vote
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### Can we put bounds on the number of stationary points over parameter space for a vanilla MLP where loss is MSE?

Suppose $f(\vec{x}; \vec{\theta}, A)$ is a fully-connected vanilla (i.e. alternating matrix multiplications and sigmoid functions) multilayer perceptron that takes a vector $\vec x$ from the features ...
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### Making intuition about the Hamilton principle in classic mechanics?

I am trying to develop intuition about Why happen to be true the Hamilton's principle of stationary action, and after seen this video I have a few questions that are more related to maths than physics....
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1 vote
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### Calculate local minima and maxima

Consider the function $f : \mathbb{R^2} → \mathbb{R}$ defined by $$f(x_1, x_2)=e^{-(2x_1^2+3x_2^2)}$$ Determine whether the stationary point is a strict local maximum or a strict local minimum.  ...
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### Calculate stationary points and local maxima and minima

Let $a, b∈ \mathbb{Z}$ \ {$0$} and let $f: \mathbb{R^2}→\mathbb{R}$ be defined by $$f(x_1, x_2)=ax_1^2+bx_2^2-4ab^2x_1-2a^2bx_2.$$ Find all stationary points of $f$ and, if possible, determine which ...
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1 vote
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### Proof for stability of stationary point

Given a twice continuously differentiable function $f$ used for a difference equation $x_{n+1} = f(x_n)$, we can show that a stationary point $f(s) = s$ is asymptotically stable (see e.g. here for ...
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830 views

### What is the condition for the saddle point of a function of three variables?

For a function $f(x,y)$ of two real variables $x$ and $y$, a point $(x_0,y_0)$ is a saddle point if the determinant of the Hessian matrix $$[f_{xx}f{yy}-(f_{xy})^2]_{x=x_0,y=y_0}<0.$$ If we are ...
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1 vote
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### 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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1 vote
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### 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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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 ...
### 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 ...