1
vote
1answer
40 views

What to do *rigorously* when the second derivative test is inconclusive?

How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive? Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions? ...
-1
votes
1answer
21 views

Application of a derivative [closed]

A spherical projectile 40 cm in diameter and weighing 32kg is shot directly upward from ground level at 196m/sec. Ignoring air resistance during its flight, what is the max height the ball will ...
0
votes
0answers
28 views

how to find the optimal function with lagged cost? (calculus of variations)

I need to find the function $b( )$ that maximizes this guy ($c()$ and $\beta()$ are functions too, and $c()$ is convex): $$\int_{0}^{T} \! e^{-\delta v}\beta(v) \left[\int_{0}^{v} b(s) \; ds - ...
1
vote
2answers
33 views

How to find local maximum of the function $f(x) = x^3-9x^2+24x+4$?

Give the value of x where the function $f(x) = x^3-9x^2+24x+4$ has a local maximum. a) -4 b) 4 c) 2 d) 3 e) -2 I graphed it and I'm not sure how to find the local max
6
votes
3answers
62 views

Interesting, unusual max/min problems?

So I've got to that stage of my elementary mathematics subject for engineers when we talk about differentiation and solution of max/min problems. And I'd like to entertain and engage the students ...
0
votes
1answer
40 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
0
votes
0answers
25 views

Sufficient conditions for the objective function to have gradient pointing towards the origin

Say I have a sufficiently smooth objective function $J(x)$. How can I ensure the below statement is fulfilled -with additional assumptions to $J$ if needed. $\underset{{{| x ...
3
votes
1answer
66 views

Computing the derivative of a transformation matrix

I am trying to find a geometric transformation between two images, where the transformation is a simple scaling matrix. So, if I denote the two image functions as $r$ and $f$ and the scaling matrix as ...
2
votes
4answers
83 views

Find the point where the function $C(t) = 0.24t/(t^2 + 5t +4)$ attends its maximum [closed]

The concentration, $C$, in micrograms per deciliter, of a drug in a patient's bloodstream is given by the equation, $C(t) = \dfrac{0.24t}{t^2 + 5t +4}$ , where $t$ is the number of hours after the ...
0
votes
2answers
53 views

Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
0
votes
0answers
24 views

Differentiation of cost function in adaptive CFO estimator

I'me trying to simulate the steepest descent algorithm for CFO estimation using null subcarriers (OFDM wireless). And some mathematic difficulties have arised. In the core of algorithm lies cost ...
1
vote
2answers
62 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
1
vote
1answer
21 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
4answers
87 views

Local minimum of $f(x) = 4x + \frac{9\pi^2}{x} + \sin x$

What's the minimum value of the function $$f(x) = 4x + \frac{9\pi^2}{x} + \sin x$$ for $0 < x < +\infty$? The answer should be $12\pi - 1$, but I get stuck with the expression involving both ...
0
votes
0answers
32 views

Deriving stationary points using the second order derivative.

Suppose that for some function $f$ we want to know the stationary points, i.e. $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$. We can define a new function ...
1
vote
2answers
75 views

Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
0
votes
1answer
55 views

$y(x) = \int_0^x \frac{\sin(t)}{t}dt $

Let $y(x) = \int_0^x \frac{\sin(t)}{t}dt $ find maximums and minimums of $y(x)$. First let $F(x) = \int_0^x \frac{\sin(t)}{t}dt$ and $f(t) = \frac{\sin(t)}{t}$ then $F'(c) = f(c) $ then if $ ...
1
vote
3answers
53 views

Calculate minimum perimeter of a rectangle with an extra constraint.

I have been set this problem, and although I can derive a minimum perimeter using calculus, I now need to add an extra constraint to one side of the rectangle and I am having problems deriving a ...
1
vote
0answers
27 views

Local extrema given the graph of a function's derivative

I am given a graph of the derivative of a function and answered most of the questions, but am still stuck at answering where the local extrema are. I had a sample question to reference from and it ...
0
votes
0answers
21 views

How to find the minima for $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$?

Please guide in how to find the value of $x$ for which $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$ will be minimum. I know this involves differentiation but am not sure on how to ...
0
votes
4answers
61 views

The sum of two variable positive numbers is $200$. Find the maximum value of their product.

The sum of two variable positive numbers is $200$. Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$. NB: I'm currently on the ...
2
votes
2answers
54 views

Finding the absolute minimum and maximum of a function

The function is $$f(x)=x+\sin(2x)$$ I need to find the absolute maxima and minima of several different domains using this function. I have found that the derivative of this function is ...
0
votes
0answers
35 views

Maximizing sum with a constraint

Given the function $$ f(\alpha_{1},\ldots,\alpha_{k})=C\sum_{i=1}^k \alpha_i e^{-(b^2/d)\alpha_i}\text{ with } C>0,\ b>0,\ d>0,\ \forall i\in\{1,\ldots,k\}:\alpha_{i}\ge 0 $$ with the ...
1
vote
2answers
51 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
0
votes
1answer
17 views

Finding coefficients of a third degree polynomial

The third degree polynomial $$-x^3 + ax^2+bx+c$$ has an maximum at $(2,10)$ and an inflation point at $(0,-6)$. Find the coefficients $a$ $b$ and $c$. Am I supposed to differentiate the polynomial ...
0
votes
2answers
49 views

fence a circular land and a square land.

With a wire mesh of 1000 mts divided into two parts , we want to fence a circular land and a square land. a)Calculate the lengths of each of the parties such that the total area enclosed is ...
0
votes
1answer
28 views

Find f with A plane curve whose equation is $y - f (x) = 0$ passes through the origin.

A plane curve whose equation is $y - f (x) = 0$ passes through the origin.Consider the rectangle $R_x$ formed by the coordinate axes and lines parallel to the axis passing through the point $(x, f ...
1
vote
1answer
46 views

A particle traveling with velocity $v_a $ in the medium $A$ and with velocity $v_b $in the medium $B$

A particle traveling with velocity $v_a $ in the medium $A$ and with velocity $v_b $ in the medium $B$. The particle starts at time $t = 0$ from the point $P_i$ and has to get in the minimum time to ...
3
votes
2answers
214 views

Two halls 6 and 9 meters perpendicularly intersect. Optimization

Two halls 6 and 9 meters perpendicularly intersect. Find the length of the longest straight bar to be passed horizontally from one aisle to another by a corner without deformation. and this is my ...
1
vote
1answer
35 views

First derivative test and uniqueness of local extrema

This is the context in which my question lies. See below for the actual question. Let $f(x)$ be differentiable everywhere and have a minimum at $x^*$. Then for every $x$ in a proper neighbourhood ...
0
votes
1answer
44 views

How can I find the vertices of a triangle by optimization?

Here is the information provided, and the hypotenuse length is minimum. How can I find the vertices of a triangle by optimization? Thanks.
0
votes
4answers
64 views

Find the absolute maximum/minimum values of S(t) where S'(t) is a quartic function with lots of horrible decimal places.

So I have a problem where I'm to find the absolute maximum and minimum values of the following function... $S(t) = -0.00003237t^5 + 0.0009037t^4 - 0.008956t^3 + 0.03629t^2 -0.04458t + 0.4074$ ...
1
vote
1answer
91 views

Can Moore–Penrose pseudoinverse solve for underdetermined linear system?

Thanks for reading my thread. I am thinking, many of us know that Moore–Penrose pseudoinverse can solve for overdetermined system $Ax=b$, where $x=(A^TA)^{-1}A^Tb$; for exmplae the linear regression ...
0
votes
1answer
27 views

Convex Subset Projection

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
2
votes
3answers
96 views

Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
0
votes
1answer
62 views

The minimum distance from the circle $x^2+(y+6)^2=1$ to parabola $y^2=8x$?

What are the coordinates of the points on the parabola $y^2=8x$ which are at the minimum distance from the circle $x^2 + (y+6)^2=1$?
0
votes
4answers
122 views

How to find the minimum value of $x^2+y^2+xy-4$ where $x+y=2$. [closed]

How to find the minimum value of the expression: $x^2+y^2+xy-4$ where $x+y=2$
2
votes
1answer
43 views

How exactly do I 'see' the function I need to make for optimization?

Optimization problems in Calculus seems to be my white whale. I always seem to struggle with it. I know that once I find the function I need to manipulate with it's pretty much smooth sailing from ...
0
votes
0answers
20 views

Second Frechet derivative: is there a mistake?

Let $x = (x_1, x_2, \ldots) \in l_2$ and $J(x) = \sum_{i = 2}^{+\infty} x_{i- 1}x_{i + 1}$ Calculate the first and second Frechet derivatives. Attempted solution First, let's notice that we can ...
0
votes
0answers
23 views

Derivative of the Total Variation of a digital image?

Thanks for your time reading my thread. I am trying to calculate the derivative of the Total Variation (TV) of a digital image with respect to its gray-scale intensity. Say, there is an image: ...
1
vote
0answers
25 views

Finding 'closest' function subject to constraints on derivatives

Suppose I have a real-valued function $f(t)$ for $t\in[0,T]$ s.t. $f'''(t)$ is defined as piecewise constant values: $$ f'''(t) = \begin{cases} k_0, & 0 < t \le t_0 \\ k_1, & t_0 < t ...
0
votes
1answer
145 views

Calculus Optimization - Finding the minimum cost

In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land situations. A pipeline comes to a river that is 1 km wide at point A and must be ...
0
votes
0answers
23 views

How to minimize values in equations?

If you have the equation $-a \leq \cos(45+d) \leq a$ where $a=\sqrt{\frac{(a+b)^2}{2} + c^2}$ and $(a,b,c)$ is a unit vector. for some $d$, how can you minimize $|d|$ so that the above equation is ...
0
votes
2answers
121 views

Real estate problem - local maxima

A real estate office manages $50$ apartments in a downtown building. when the rent is $\$900 $ per month, all units are occupied. for every $\$25 $ increase in rent, one unit becomes vacant. on ...
3
votes
1answer
61 views

Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
0
votes
1answer
65 views

How do I properly set up this optimization equation?

So I've been the given the task to fully optimize any packaging. I chose a DS game box. So first I took the measurements of the cartridge itself ($3.5 \text{ cm} \times 3.3 \text{ cm} \times 0.38 ...
1
vote
1answer
36 views

What does the adjoint operator do? Is this Frechet derivative correct?

Problem statement Let $x \in l^2$ and $J(x) = \sum_{n = 1}^{+\infty} x_{2n - 1}^2$ Find first and second Frechet derivatives. Attempted solution Let's note that $J(x) = \sum_{n = ...
0
votes
1answer
40 views

Frechet derivative of double integral.

Problem statement Let $u(t) \in L^{2}(0, 1)$ and $J(u) = \int_0^1 tu(t) \int_0^t u(s)dsdt$ Compute first and second Frechet derivatives. Attempted solution $$ \begin{split} J(u + h) - J(u) &= ...
0
votes
1answer
62 views

What trick to calculate this Frechet derivative?

Let $u(t) \in L^{2}(0, 1)$. I need to calculate the first and second Frechet derivatives of $$J(u) = \int_0^1 \left(\int_0^{t^3}u(s)ds\right)^2dt$$ I am completely at a loss here: I know several ...
0
votes
3answers
86 views

Linear Programming and differentiation, why can't we differentiate to find the optimum solution?

I do understand that differentiating a linear function (for a maximization) subject to some linear restriction (such as the problem $p=ax+by$ s.t. $cx+dy \leq m$) won't necessarily give me the right ...