1
vote
0answers
54 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
1answer
28 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
1
vote
2answers
33 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
-1
votes
1answer
20 views

Nearest and farthest point from a function to another [closed]

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
1
vote
2answers
36 views

Finding extremal values on a set

Let $f(x,y)=(x-1)^2+y^2+xy$. Find the maximal and minimal values of $f$ on the set $M=\{(x,y):|x|+|y|\leq4\}$. Attempt: By taking partial derivatives and solving the homogenous algebraic system we ...
4
votes
1answer
40 views

Local minimum and gradient [duplicate]

But the proof here below is specially elegant. Is there any function $f$ such that $f$ has a local minimum at $x$ but $\nabla f(x) \neq 0$? Only assumption on $f$ is that it has to be differentiable ...
3
votes
1answer
130 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
1
vote
1answer
31 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
1
vote
1answer
20 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
1
vote
0answers
26 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
0
votes
1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
0
votes
0answers
27 views

Proof that feature normalization cause faster convergence of gradient descent

How to prove that if I do feature normalization (scaling of the $x_1,\ldots,x_n$ to be all in range $[0,1]$) to a convex function $f(x_1,\ldots,x_n)$ that returns real scalar, then gradient descent ...
0
votes
0answers
12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
0
votes
3answers
40 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
1
vote
1answer
39 views

extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
0
votes
1answer
60 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
1
vote
1answer
37 views

Use Lagrange Multipliers to determine max and min

Using Lagrange Multipliers, determine the maximum and minimum of the function $f(x,y,z) = x + 2y$ subject to the constraints $x + y + z = 1$ and $y^2 + z^2 = 4$: Justify that the points you have found ...
0
votes
1answer
21 views

Gradient descent with adaptive learning ratio.

I have a neural network, trained with SGD (stochastic gradient descent) with learning ratio $\alpha$. Each iteration I try to recalculate the weights with a rule: $$\Delta \vec{w} = -\alpha ...
0
votes
1answer
46 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
1
vote
1answer
49 views

Maximum distance from the origin to the surface

I am having trouble getting the maximum distance from the origin to the surface $$ \frac{x^4}{16} +\frac{y^4}{81} + z^4 = 1 $$ Knowing I have to maximize $x^2 +y^2+ z^2$ and that the constrain ...
1
vote
1answer
27 views

Optimization of a Sum of Variables

Let there be variables $A$, $B$, $C$, $D$, and $E$ such that a total of $N$ points is allocated among the variables: $A$+$B$+$C$+$D$+$E$=$N$, $N$āˆˆ$ā„$. Let the corresponding point values returned by ...
1
vote
0answers
42 views

How to solve Max under an integral?

This is the first time I come accross a Max function inside an integral. I have looked around online and did not find anything about it. I would like to know the rules of what can I do when I have an ...
2
votes
3answers
57 views

Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers

Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers. So I set it up: $$ 1 = 2x\lambda_1 + 2\lambda_2 \\ 1 = -2y\lambda_1 \\ 1 = \lambda_2 $$ Plug ...
1
vote
1answer
29 views

Finding extrema with multiple constraints without Lagrange multipliers

Find the maximums and minimums of $z = 15x+14y$ with constraints $0 \leq x \leq 10, 0 \leq y \leq 5, 3x+2y \geq 6$ I obviously can't take the partial derivatives of inequalities, so I'm at a loss ...
2
votes
3answers
49 views

Find max/min of $e^{2x}\left(x+y^{2}+2y\right)$

$$e^{2x}\left(x+y^{2}+2y\right)$$ FOC: $$\begin{cases} 2e^{2x}\left(x+y^{2}+2y\right)+e^{2x}=0\\ e^{2x}\left(2y+2\right)=0 \end{cases}\rightarrow\begin{cases} x=\frac{1}{2}\\ y=-1 \end{cases}$$ SOC: ...
2
votes
1answer
82 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
1
vote
1answer
32 views

Locally minimizing a concave function

What will happen if we minimize a concave function via gradient descent? Where does it get stuck? Intuitively a concave function has more structure than an arbitrary function, and seem to be easier ...
2
votes
2answers
53 views

Point on $z = \frac{1}{xy}$ closest to origin

Where $x>0$ and $y>0$. I want to work with the square of the distance formula from the origin, so I went with $f(x,y) = x^2 + y^2 + \frac{1}{(xy)^2}$. Then I found the first partial ...
2
votes
1answer
80 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
2
votes
2answers
73 views

Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...
0
votes
2answers
80 views

Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...
1
vote
1answer
49 views

Absolute extrema of a multivariable function bounded by an ellipse

I have a function $f(x,y) = 2x + x^2 + y^2$ bounded by the ellipse $x^2 + 4y^2 \leq 24$ I know how to determine the extrema within the ellipse by getting the partial derivatives and setting them to ...
3
votes
3answers
60 views

Difficult time finding critical points using Lagrange

The function is $f(x,y,z) = xyz$ on $x^2 + y^2 + z^2 = 1$. So I have: $yz = 2x \lambda \\ xz = 2y \lambda \\ xy = 2z \lambda \\ x^2 + y^2 + z^2 = 1$ I guessed $x = \pm 1, y = 0, z = 0, \lambda = 0$ ...
0
votes
1answer
28 views

Maximize/minimize $1/3 x^3 + y$ with constraint $x^2 + y^2 = 1$?

I keep running around in circles when I use the Lagrangian multiplier method getting $x = 1/y$ But then when I substitute $(1/y)^2 + y^2 = 1$ I then get $1/y^2 + y^2 = 1$ and this doesn't give me ...
0
votes
1answer
46 views

lagrange multiplier (with variables x,y,z)

I'm new to this topic, pls can I get hints on how to solve it: Find the point $(x,y,z)$ obeying $g(x,y,z)=2x+3y+z-12=0$ for which $f(x,y,z)=4x^2+y^2+z^2$ is minimum. Thanks in advance.
0
votes
1answer
37 views

Finding/approximating 2 unknowns using one equation

Iā€™m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
1
vote
1answer
26 views

Understanding optimization on non-compact region

Say we have $f(x,y) = x^2 e^{-x^2 - y^2}$ and we want to optimize it over $\mathbb{R}^2$. The minimum value is $0$ since $f(x,y) \geqslant 0$; the question is whether a maximum value exists or not. ...
0
votes
2answers
40 views

Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
0
votes
3answers
62 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
1
vote
3answers
80 views

Optimize function on $x^2 + y^2 + z^2 \leq 1$

Optimize $f(x,y,z) = xyz + xy$ on $\mathbb{D} = \{ (x,y,z) \in \mathbb{R^3} : x,y,z \geq 0 \wedge x^2 + y^2 + z^2 \leq 1 \}$. The equation $\nabla f(x,y,z) = (0,0,0)$ yields $x = 0, y = 0, z \geq 0 $ ...
1
vote
0answers
24 views

Optimization non-compact region

I've unsuccessfully been looking all over the web for examples on optimizing multivariable, real-valued functions over non-compact regions. As I've understood it, such optimizations are essentially ...
0
votes
2answers
40 views

Maximum and Minimum temperature on a disc

I have a question which asks me to find the highest and lowest temperatures on a metal plate of radius 5, the temperature at point (x,y) is T(x,y)=4x^2-4xy+y^2 When I take partial derivatives of T ...
0
votes
1answer
28 views

Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
1
vote
1answer
50 views

Lagrange multiplier - Find maximum on surface

I need someone to walk me through a 3 variable lagrange problem, since I haven't been able to find a reliable source to teach me, please. Here it is: Find the maximum of the function $F(x,y,z) = ...
3
votes
4answers
63 views

Optimization with a constrained function

Okay so I understand how to find points of extrema when for example, We have $3x^2 + 2y^2 + 6z^2$ subject to the constaint $x+y+z=1$. I followed the method of the Lagrange multiplier and resulted in ...
0
votes
0answers
18 views

Economics - Function satisfying two conditions

A firm has a production function $y=f(x_1, x_2)$; that is, for specific level of inputs $x_1, x_2$, the total output of the firm is determined. The costs of the inputs are $p_1, p_2$ respectively, ...
1
vote
1answer
35 views

$f:R^2\to R$: Determining the Nature of a Critical Point when the Second Derivative Test Fails

I'm reviewing for a final exam tomorrow. This is an exercise that I am having trouble with: The function: $f(x,y)=x^2-y^4$ I determined that there is one critical point, at $(0,0)$. I determined ...
0
votes
2answers
41 views

Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
0
votes
2answers
74 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
0
votes
1answer
24 views

Finding minimum norm

Let $A$ be $k\times k$ positive symmetric matrix, $K$ is $k\times d$ full rank matrix with $d<k$, and $v\in\mathbb{R}^k$. I'd like to find $x\in \mathbb{R}^d$ such that $(Kx-v)^TA(Kx-v)$ minimum. ...