All Questions
Tagged with hessian-matrix maxima-minima
64 questions
0
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1
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40
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Show that a harmonic function has no local maxima and minima
Given a differentiable function $f: U \rightarrow \mathbb{R}$, a point $a \in U$ is called a critical point of $f$ (or a singular point) when $d f(a) = 0$, that is,
$$
\frac{\partial f}{\partial x_1}(...
- 2,413
0
votes
3
answers
72
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Determining the minimum value of the function $f(x, y) = \frac{x}{y} + \frac{y}{x}$ with $x, y\in \mathbb{R}_{>0}$.
I want to conclude that the minimum value of the function $f(x, y) = \frac{x}{y} + \frac{y}{x}$ with $x > 0, y > 0$ is 2 obtained along the line $x = y$, but I seem to be a bit rusty with my ...
- 1,321
0
votes
1
answer
49
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This function has no saddle points: correctness of this reasoning
I would like to know if my reasoning is correct. I have the function $f(x, y) = e^{3x}(1+25x^2+25y^2)$ and I have to study the stationary points.
After computing the gradient I found
$$\begin{cases} 3 ...
- 3,521
0
votes
0
answers
29
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What is the intepretation of a Hessian with only negative and zero eigenvalues?
I'm looking at a high-dimensional optimisation problem. Specifically, one involving a function $F(c_n)$ which depends on N parameters (in my case $N=200$). I have been able to minimise this function ...
0
votes
1
answer
51
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Classifying critical points when hessian has det0 [closed]
I am trying to find the critical points of
$$f(x,y)=(x^2-y^4)(1-x^2-y^4).$$
There are 9, and I found all their coordinates. I’ve classified 6/9 using the hessian, but the remaining 3 have at least one ...
- 69
-1
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1
answer
34
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k-Jet of a function $f:R^3 \rightarrow R$
I am studying this article. Can someone explain to me why the 4-jet of the function
$U(x, y, z) = x^4 + y^4 - z^6$
is equal to
$(j^{(4)} U)(x, y, z) = x^4 + y^4$
Also, the author says that the origin ...
- 165
0
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0
answers
381
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Critical points with Hessian determinant equals to zero
Consider the function $$f(x, y) = x^4+y^4-y^2$$
After having compute the gradient, I found the following critical points
$$p_0 = (0, 0) \qquad \qquad p_{1, 2} = \left(0, \pm \sqrt{\frac{1}{2}}\right)$$...
- 3,521
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0
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134
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Prove that if a function has a Hessian with zeroes on the diagonal, the function doesn't have local maxima
For a function who's Hessian in the form
$$\begin{pmatrix}
0 & a & b\\
a & 0 & c\\
b & c & 0\\
\end{pmatrix}$$ prove that the function can not have any local maxima.
I tried to ...
- 1
1
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1
answer
250
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Second partial derivative: What happens if $f_{xy}^2$ is larger than $f_{xx} f_{yy}$
Considering $(a,b)$ is a critical point of a funciton $f(x,y)$ and
$D(x,y) = det(H(f(x,y))) =f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2}$ is the determinant of the hessian matrix from that function.
If ...
- 11
4
votes
1
answer
106
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Extrema of a surface $z=f(x;y)$ when $det(H)=0$
I'm given the following problem:
$\text{Examine}\ z=f(x;y)=x^4+y^4+18xy-9x^2-9y^2+1\text{ for extrema and saddle points.}$
It is trivial to find $\nabla f=(4x^3+18y-18x; 4y^3+18x-18y)$ and the ...
- 364
0
votes
1
answer
543
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Prove the existence and uniqueness of a global minimum for a quadratic function $q(x)$ with positive definite Hessian
How do I prove the existence and uniqueness of a global minimum for a quadratic function $q(x)$ with positive definite Hessian?
1
vote
0
answers
152
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Max, min or saddle point?
Consider the function on $\mathbb{R}^2$:
\begin{equation*}
V(x,y) = (x^2+y^2)^2-a(x^2+y^2)+a^2/4
\end{equation*}
with $a$ real. I want to decide weather the critical points are max, min or saddle ...
- 297
1
vote
1
answer
716
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Find the critical points of a multivariable function with constraints
For my vector calculus class, I need to solve this problem:
Let $$f(x,y)=x^3+8xy-8y^3-7x+14y$$
Given that this function has critical points on the lines $3x=7$ and $3x=-3$, find and classify all ...
- 346
0
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0
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60
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Checking for local extrema of two variable functions
Let $f:\mathbb{R}^2\to\mathbb{R}$, $f(x,y)=xye^{-(x+y)}$.
I want to check for local extrema and saddle points.
The gradient is given by:
$\nabla f(x,y)=((y-xy)e^{-(x+y)}, (x-xy)e^{-(x+y)})$
Solving ...
- 11.3k
2
votes
1
answer
213
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Type of critical points in three dimensions
I am facing an exercise about maxima and minima for the function
$$f(x, y, z) = xye^x - xyz$$
So the gradient is
$$\nabla f(x, y, z) = (ye^x + xye^x - yz, xe^x - xz, -xy)$$
The solutions I found are ...
- 3,521
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1
answer
252
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Cubic equation in the Hessian polynomial
I am stuck in solving the cubic equation which does arise from the calculation of the characteristic polynomial of a Hessian matrix.
Starting from the beginning, the problem is a max-min problem:
$$f(...
- 3,521
2
votes
1
answer
247
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Find the extrema of $f(x,y) = x^3\cdot y^3$ in $\mathbb{R}^2$
I am asked to find the extrema of
$$f(x,y) = x^3\cdot y^3$$
in $\mathbb{R}^2$
However, using the Hessian criteria, I get that the determinant of the Hessian matrix is zero for the two possible ...
- 23
1
vote
0
answers
151
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Is $f(x, y) = xy-2x-y$ an hyperbolic paraboloid?
so I solved the function, finding a hyperbolic paraboloid with a saddle point $xy-2x-y=-2$ at $(x, y)=(1, 2)$, with no global or local maxima and/or minima. But the answer sheet marks that it indeed ...
- 11
0
votes
1
answer
82
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There is a value less than Global minimum!
Here we see that minimum of the function $x+y+1/xy$ is obtained at $(1,1)$ which is $3$
However , $f(1,-1) = -1$. Then how is $f(1,1)$ actually the minimum?
1
vote
2
answers
261
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Find the sign of eigenvalues generated from Hessian matrix
I have to classify a critical point of a function of 3 variables. I determined the Hessian and I know I can orthogonally diagonalise this, so to classify the critical point, I only need to know the ...
- 515
1
vote
1
answer
171
views
Maxima and minima of quadratic function via the Hessian
Find maxima and minima of the quadratic function $f: \Bbb R^2 \to \Bbb R$ defined by $$ f(x_1,x_2) = x_1 (x_1 - 1) x_2 $$
I found the critical points to be $P_1(0,0)$ and $P_2(1,0)$. I know $P_2$ is ...
- 123
2
votes
1
answer
76
views
Potential function with a fixed set of minima and no poles.
Can we find a continuous, almost-everywhere smooth, potential function with no poles, such that the set of local minima for the sum of these potentials around $n$ points are exactly those $n$ points?
...
- 962
4
votes
1
answer
244
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Hessian of restriction of a map to the boundary of a domain
Let $(M^n,g)$ be a Riemannian manifold and let $\Omega \subset M$ be a smooth and bounded domain of $M$. Suppose $u : \overline{\Omega} \to \mathbb{R}$ is a smooth function satisfying both $u = 0$ and ...
- 5,235
3
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1
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207
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Hessian at a maximum point lying on the boundary
Let $\Omega$ be a bounded domain of class $C^2$ in $\mathbb{R}^n$ and let $f: \overline{\Omega} \to \mathbb{R}$ be a smooth function. Assume $f$ attains its maximum at $x_0 \in \partial \Omega$. Can ...
- 5,235
0
votes
1
answer
252
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Multivariable Non-degenerate Critical Points Question
Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a $C^2$ function, and the origin is a non-degenerate critical point and suppose $f(x,mx)$ is a local minimum at the origin for all $m$, then does $f$ ...
- 1,149
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0
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290
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Sufficient condition for saddle point
Consider a smooth function $F : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ with $m,n > 1$.
Definition: $(x^* , y^*) \in \mathbb{R}^n \times \mathbb{R}^m$ is a saddle point of $F$ if there ...
- 1,170
0
votes
1
answer
186
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Multivariable Function Optimization with Semidefinite Hessian Matrix
I'm writing to ask for support in carrying out an exercise about the optimization of a multivariable function.
The function is: $f(x,y,z)=x^2+y^2z+z^2-2x$
Clearly the domain is $\mathbb{R}^3$, and $f$...
- 1,584
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4
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1k
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Why do we need to determine the definiteness of the Hessian to decide what a critical point is?
In univariate calculus, if we know that $f'(c)=0$, we can determine if the function $f$ has a minimum at $c$ by checking that $f''(c) > 0$. The multivariate analogue of the second derivative is the ...
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0
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1
answer
253
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Local stability of a min-max point
Suppose I have a smooth scalar function $f(x,y)$ where $x,y$ can be vectors. I am interested in finding a saddle-point:
$$\min_x \max_y f(x,y)\qquad(1)$$
I have the intuition that at such a point, ...
- 6,891
0
votes
1
answer
36
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Multivariable Calculus question, critical points and boundness.
I'm doing an exercise from the book Multivariable Real Analysis by Kolk and Duistermaat, however I'm not sure how to proceed after some point. It asks the following: we have $f:\mathbb{R}^{2}\...
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-2
votes
1
answer
28
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Finding the local minima,maxima and saddle point of $f(x,y)=y^2e^{-x}-x^2$
I did this question and got one critical point $(0,0)$. Whilst solving it using Hessian Matrix method, the $D(0,0)$ was found as $0$ which makes it inconclusive. However,I am having doubts on the way ...
- 11
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votes
1
answer
1k
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When the Hessian determinant is zero, the second derivative test is inconclusive. Why?
We have if $(a,b)$ is a critical Point, $f_{xx}\ne 0$ and:
$$f_{xx}f_{yy}-f^2_{xy}=0$$
By Taylor's Formula and using Operator notation we get:
$$f(a+h,b+k)=f(a,b)+\left(h\frac{\partial }{\partial x}...
- 13.3k
0
votes
1
answer
98
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Local Maxima/ Minima of $F(x,y)=(x^2+y^4-1)^2$
Searching for Local Minima/ Maxima of the function $F(x,y)=(x^2+y^4-1)^2$ I found out that $(0,0)$ and $(\pm \sqrt{1-y^4},y)$ are possible candidates. Looking up in Wolfram-Alpha that seems to be ...
user695468
4
votes
1
answer
370
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A Problem from Past Entrance Exams into Math Master's Program in Taiwan
Let $f$ and $g$ be $\mathbb{R}$-valued $C^{\infty}$ functions on $\mathbb{R}^2$ and let $S=\{(x,y)\in\mathbb{R}^2|f(x,y)=0\}$.
Suppose that at some point $p=(a,b)\in S$ we have
$\frac{\partial f}{\...
user695163
0
votes
2
answers
260
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Does $F1(x,y) = \frac{1}{4}x^4 + x^2 y + y^2$ has a minimum through positive definite of the second derivative matrix?
I took this question from the book Introduction to Linear Algebra (Gilbert Strang). It is from section 6.5, exercise 29:
For $F1(x,y) = \frac{1}{4}x^4 + x^2 y + y^2$ and $F2(x,y) = x^3 + xy - x$ ...
- 185
1
vote
1
answer
869
views
Extreme value verification using Bordered Hessian matrix
Determine the minimum value of $$bcx+cay+abz$$ subject to the conditions $$xyz=abc$$
Here is the solution in the book (using the Lagrange multipliers)
$$F = bcx+cay+abz+k(xyz-abc)$$
After solving ...
- 335
4
votes
0
answers
308
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Strategies for proving that a critical point is neither maximum nor minimum
Let $f(x,y) = x^3 + y^3 -3x$. Since it's everywhere differentiable, we know that its only critical points are $(1,0),(-1,0)$ given that the gradient $\nabla f(x,y) = (3x^2-3,3y^2)$ is zero if and only ...
- 781
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votes
1
answer
150
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Show that the critical point of the function is a local minimum.
Consider the function $F:\mathbb{R}^2\to\mathbb{R},(x,y)\mapsto F(x,y)=x^2+2y^2+4.97$.
$H$, hessian matrix of above function is symmetric. If $H$ is positive definite in a critical point then it is a ...
- 111
0
votes
1
answer
181
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Find the global and local extrema in set
Given the function: $f(x, y) = x^2 - y^2$
Determine the local and global extrema for f in the set $M = \{(x, y) \in \mathbb{R^2}\ |\ y + e^{-x^2} -1 =0\}$
I know about Lagrange Multipliers, but we ...
- 455
2
votes
2
answers
2k
views
Classifying the stationary points of $f(x, y) = 4xy-x^4-y^4 $
$f(x, y) = 4xy-x^4-y^4 $
The gradient of this function is $0$ in $(-1, -1), (0, 0),(1, 1)$
I tried to compute the determinant of the Hessian Matrix, but it's 0 for every point, I always get a null ...
- 129
0
votes
1
answer
61
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Multivariable calculus - Trying to understand if a stationary point is a saddle point, max or min
I have the following function
$f(x, y) = x^4+y^4 $
$(0, 0)$ is a stationary point, so I calculat the determinant of the Hessian Matrix, which is 0, so I try to understand what kind of point that is ...
- 129
1
vote
1
answer
628
views
Prove that function has minimum when Hessian is zero
Prove that the following function has a minimum at $(0,0)$.
$$ f(x,y) = \dfrac{1}{24}[e^{2y+2\sqrt{3} x}+e^{2y-2\sqrt{3} x} + e^{-4y}] - \dfrac{1}{8} $$
My attempt:
I tried to solve it via hessian ...
- 816
0
votes
1
answer
567
views
Bordered Hessian matrix to find a minimum of the function
I was trying to find the global minimum for the function $$(a + b) z + (a + c) y + (b + c) x $$ subject to the following constraint: $$(xy + xz + yz)(ab + bc + ac)=1.$$ By Lagrange multipliers I found ...
2
votes
2
answers
125
views
Minimum of $x_1^2+x_2^4$
I am asked to find a minimum of $f(x_1, x_2) = x_1^2+x_2^4$ applying the optimality conditions.
I am stuck. What I have found is not conclusive. I show what I have done:
Testing the First-Order ...
- 349
0
votes
1
answer
56
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How to study the critical points of a $2$-variable function?
I am revising some past exam questions and there is one that states:
Study the critical points of the function:
$$f(x,y)=x^2+y^2-x^4-y^4-2x^2y^2.$$
According to my professor, this is what I have ...
- 137
0
votes
1
answer
3k
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Using the Hessian Matrix to classify points
From what I've gathered from my calculus supplements and the web, I want to know if I have the general computation procedure understood correctly.
Example: Given f such that f(x,y) = ___. Find and ...
- 53
0
votes
1
answer
281
views
Local minimum of an analytic function
This is a follow-up to a previous question of mine. I know that any local minimum $x_0$ of a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. ...
- 1,824
3
votes
2
answers
711
views
Local minimum has neighbourhood with positive semi-definite Hessian?
I know that any local minimum $x_0$ of a (twice continuously differentiable) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. Can we say ...
- 1,824
0
votes
1
answer
71
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Evaluating an extremal value if the hessian matrix has at least one eigenvalue which is zero
$$f(x,y) = 2x^4-3x^2y + y^2$$
We want to find all extremal values:
$$df(x,y)=(8x^3-xy,-3x^2+2y)\overset{!}{=}0 \quad \Rightarrow \quad p=(0,0)$$
$$H_f(x,y)=\begin{pmatrix}24x^2-6y& -6x\\-6x &...
- 897
1
vote
0
answers
78
views
Classifying a degenerate point of a function of three variables
Classify the critical point (1,$\frac{\pi}{2}$,0) of the function $f(x,y,z) = xsinz-zsiny$
I found the Hessian matrix but it had a zero determinant.
I tried looking at how the function behaves when ...
- 33