What is Extrema: Definition and 153 Discussions

Photedes extrema, the concolorous, is a moth of the family Noctuidae. The species was first described by Jacob Hübner in 1809. It is found in most of Europe (except Iceland, Ireland, the Iberian Peninsula, Norway, Italy, Bulgaria and Greece).

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1. MHB Extrema and convergence of sequence

Hey! :giggle: For $n\in \mathbb{N}$ let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ given by $f_n(x)=\frac{x+2n}{x^2+n}$. (a) Determine all (local and global) extrema of $f_n$ and the behaviour for $|x|\rightarrow \infty$. Make a sketch for $f_n$ and $f_n'$. Show that there exists $x_1<x_2<x_3<x_4$...
2. MHB Global Extrema of a Trigonometric Function

Hey ! :giggle: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $f(x,y)=\sin^2(x)\cdot \cos^2(y)$. - Show that $f$ has at $\left (\frac{\pi}{2}, 0\right )$ a strictly local maximum and that is also a global maximum. - Determine all points at which $f$ gets its global minimum. I have sone...
3. Finding the relative extrema for a speed function using parametric curves

I have no problem in following the literature on this, i find it pretty easy. My concern is on the derived function, i think the textbook is wrong, it ought to be, ##S^{'}(t)##=##\frac {4t} {\sqrt{1+4t^2}}=0## is this correct? if so then i guess i have to look for a different textbook to use...
4. MHB Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x

Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x So here is what I have so far: The derivative is 1-ln(x)/x^2 Critical points are (e,1/e) No concavity Local max is also (e,1/e) (no local min) no inflection points Increase on (0, e) and...
5. Finding Specific Extrema when grad(F)=constant & Lagrange Gives y=-z/2

I found that f= x -2yz. To maximize f, I can first inspect the solutions to grad(F)=0. z=y=0 pops out, but I'm not sure what to do with the x-component equaling 1. Do we just include (x,0,0) as a solution? I think the problem wants specifics though, based on what I've seen previously from...
6. Find the local extrema then classify them

So this is a very long answer for a question worth 4 marks. Would it be a good idea to keep the explanation, or to have it strictly math only?
7. I An interesting point regarding critical points and extrema

Hi all, I have recently faced some problem about distances between two curves, and (re?)"discovered" an interesting point that I would like to share with you. In the following, we consider a function of two variables ##f(x,y)##, but it should be clear that the definitions and the result is...
8. Extrema of electrical potential

Homework Statement A charge of 2 C is located at the origin. Two charges of −1 C each are located at the points (1, 1, 0) and (−1, 1, 0). If the potential φ is taken to be zero at infinity (as usual), then it is easy to see that φ is also zero at the point (0, 1, 0). It follows that somewhere...
9. Critical Points of a Parameter Dependent Integral

1. The problem statement, all variables, and given/known data Find and categorize extremes of the following function: $$F(y)=\int_{y}^{y^{2}}\frac{1}{\ln^{2}x}dx$$ for ##y>1##. Homework Equations $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...
10. MHB -z.55 Find the value(s) of t corresponding to the extrema

$\text{Find the value(s) of$t$corresponding to the extrema of}$ $$f(x,y,z)=\sin(x^2+y^2)\cos(z)$$ $\text{subject to the constraints}$ $$\text{x^2+y^2=4t, 0\le t\le\pi, and z=\frac{\pi}{4}}$$ $\text{Classify each extremum as a minimum or maximum.}$ \begin{align*} \displaystyle...
11. MHB -x4.6 Find extrema f(x, y, z) = x + yz

$\text{Find the extrema of$f(x, y, z) = x + yz$on the line defined by}$ $$\text{x = 8(2 + t), y = t - 8, and z = t+ 2.}$$ $\text{Classify each extremum as a minimum or maximum.}$ \begin{align*} \displaystyle & \text{Book answer}=\color{red}{\text{$(8, -9, 1)$, minimum}} \end{align*}...
12. I Help please with biocalculus question involving differentiation

Hi, I was just wondering how one would arrive at the answers to these questions. I have the solution for parts a and b, but not for part c. Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus...
13. MHB A text problem on the study of extrema of a function

Consider squares inscribed in different isosceles triangles with sides equal to 1. (One side of the square lies on the bottom.) Find the side of the largest square
14. MHB Do we get the extrema from that graph?

Hey! :o I am looking at the following exercise: Detremine the extrema of the function $f(x,y)=x^2y$ subject to $3x+2y=9$. Prove also the second order condition. What kind is the extremum? Is this an extremum of the whole function $f(x,y)$? Draw the contour lines of $f(x,y)$ and the...
15. MHB Finding Extrema under Constraints

Hey! :o I want to find the critical points of the function $f(x_1, x_2)=x_1x_2$ under the constraint $2x_1+x_2=b$. Using the method of Lagrange multipliers I got the following: \begin{equation*}L(x_1,x_2,\lambda )=x_1x_2-\lambda \cdot \left (2x_1+x_2-b\right )\end{equation*}...
16. MHB How to Find the Absolute Extrema of a Function Over a Defined Region?

Find the absolute extrema of the function over the region R. (In this case, R contains the boundaries.) f (x, y) = 12 - 3x - 2y R: The triangular region in the xy-plane with vertices (2, 0), (0, 1), and (1, 2). I need the steps to guide me through this monster question. I am familiar with...
17. MHB Relative Extrema vs Absolute Extrema

In basic terms, what is the main difference between relative extrema and absolute extrema? I know that absolute extrema is more involved but why is this the case?
18. MHB Critical Points & Extrema of Multivariable Function

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. f (x, y) = x^(2/3) + y^(2/3) Solution: f_x = 2/[3 (x)^1/3] f_y = 2/[3 (y)^1/3] f_xx = -2/[9 x^(4/3)] f_yy = -2/[9 y^(4/3)] f_xy = 0 I set f_x and f_y to 0 and found...
19. MHB Find Absolute Extrema Over Region R

Find the absolute extrema of the function over the region R. (In this case, R contains the boundaries.) f (x, y) = 12 - 3x - 2y R: The triangular region in the xy-plane with vertices (2, 0), (0, 1), and (1, 2). I need the steps to solve such a problem. I was told to graph the 3 given points...
20. MHB Critical Points & Extrema of f (x, y)

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. f (x, y) = x^3 + y^3 Solution: f_x = 3x^2 f_y = 3y^2 f_xx = 6x f_th = 6y f_2xy = 0 I set f_x and f_y = 0 and found the critical points to be (0, 0). Is this...
21. MHB Finding Critical Points and Extrema for g(x, y) = sqrt{x^2 + y^2 + 1}

Find the critical points and extrema of the function g (x, y) = sqrt {x^2 + y^2 + 1}. Can someone get me started here? I also would like the solution steps. I said solution steps not the solution. Do it like this: Step 1... Step 2... Step 3...etc...
22. MHB Extrema of Functions of Two Variables

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. Given: f (x, y) = (x - 1)^2 (y + 4)^2 I found the partial derivative for x and y to be the following: f_x = 2 (x - 1)(y + y)^2 f_y = 2 (y + 4)(x - 1)^2 I solved for x...
23. Extrema of Two Variable Bounded Function

Homework Statement Find the maximum and minimum value attained by f(x, y) = x2 + y2 - 2x over a triangular region R with vertices at (0, 0), (2, 0), and (0, 2) Homework Equations partial x = 0 and partial y = 0 at extrema The Attempt at a Solution partial x = 2x - 2 partial y = 2y 2x - 2 =...
24. MHB Extrema of function - What is λ?

Hey! :o We have the function $f(x_1, x_2, x_3)=9x_1\cdot x_2\cdot x_3$ and we want to find possible extremas under the constraint $2x_1+x_2+x_3=m, m>0$ and $x_1, x_2, x_3>0$. Then I have to calculate $x_1^{\star}(m), x_2^{\star}(m), \lambda^{\star}(m)$. I have done the following...
25. Where are the extrema and inflection points for the function y = x/ln(x)?

Homework Statement Find all extrema and inflection points of the function ## y = \frac {x} {ln(x)} ## Homework Equations I did the first and second derivative by hand, and they worked out in CAS as well... ## y = \frac {x} {ln(x)} ## ## y' = \frac {lnx - 1} {(ln(x))^2} ## ## y'' = \frac {2...
26. MHB A Function's Intervals of Increasing/Decreasing, Extrema and Concavity

Suppose that f(x) = (x^2 + 10)(4 - x^2). (A) Find all critical values of f.Critical value(s) = (B) Use interval notation to indicate where f(x) is increasing. Increasing: = (C) Use interval notation to indicate where f(x) is decreasing. Decreasing: = D) Find the x-coordinates of all local...
27. MHB Finding relative extrema of a cubic function

Find local maxima and minima for 6${x}^{3}$+6${x}^{2}$-8x. I found that (-1.08,8.08) is max, (0.41,-1.86) is min. Where i was wrong?
28. Absolute extrema 2 variable function

Homework Statement find the absolute extrema of f(x,y) = 2x - 2xy + y^2 in the region in the xy plane bounded by the graphs of y= x^2 and y = 1 The Attempt at a Solution first we find the first partials fx(x,y) = 2 - 2y fy(x,y) = 2y-2x 2-2y = 0 when y = 1 2y - 2x = 0 when y=x in this case...
29. Absolute extrema function of 2 variables

Homework Statement I need to find the absolute extrema of the function in the specified region f(x, y) = x^2 + xy R = {(x,y): |x|<=2, |y|<=1}The Attempt at a Solution The first partial derivatives are fx(x,y) = 2x+y and fy(x,y) = x They are both 0 only when x and y are both 0. So...
30. Find Relative Extrema of f|s: Explained with Lagrange Multipliers

I am in Calculus 3, and I do not under stand what it means when they ask to find the relative extrema of f|S? The problem is usually something like f:R^n=>R, (x,y,z) |=> (some function) , S= {(x,y) | x e R} What does f|s mean? How does this relate to Lagrange multipliers? The book does not...
31. Constrained Extrema and Lagrange Multipliers

Suppose I have a function f(x,y) I would like to optimize, subject to constraint g(x,y)=0. Let H=f+λg, The extrema occurs at (x,y) which satisfy Hy=0 Hx=0 g(x,y)=0 Suppose the solutions are (a,b) and (c,d). If f(a,b)=f(c,d) , how do I determine whether they are maxima or minima?
32. Confusion with Relative Extrema and Intervals

I've been watching the Khanacademy videos on Calculus and in this video, at 4:18: He talks about relative minima and maxima in an interval. He says that the relative extrema can't be at the endpoints. As far as I understand, in that case the interval would have to be an open one, but my...
33. Can a function have a local max but no global max?

Homework Statement I need to confirm if I correct in saying the following: If f(x) is a function having the domain [a,b) as shown in the figure, then f(x) has several local maxima but none of them is global maximum, and f(x) does not have a global maximum. Homework Equations and...
34. MHB Critical Points and Global Extrema Question

Hey guys, I'm doubting some of my answers and I'd appreciate some help. I'm only asking about 2abc, ignore 1ab please: For 1a, I simply took the derivative (as I did with all three of these questions) and calculated global and local extrema and critical points. Ultimately, I found that...
35. MHB Local Extrema of Quartic Function: Help Find (x, y) Points

Here is the question: I have posted a link there to this thread so the OP can view my work.
36. Second Derivative Test for Local Extrema

Suppose ##f^{\prime\prime}## is continuous on an open interval that contains x = c 1. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)<0##, then ##f## has local maximum at x = c. 2. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)>0##, then ##f## has local minimum at x = c. 3. If...
37. MHB Local Extrema of f(x,y)=3sin x sin y

Here is the question: I have posted a link there to this thread so the OP can view my work.
38. First-Order Extrema in Classical Mechanics , Theoretical Minimum

First-Order Extrema in "Classical Mechanics", Theoretical Minimum In the 3rd lecture of Classical Mechanics, 2011, by Dr. Susskind in his Theoretical Minimum series, he talks about calculating extrema, saddle points, etc. to "first order". "if you move a little bit, the potential is zero, to...
39. MHB Find x,y Coordinates of Stationary Point: 2x^2-2xy+y^2+2x+5

Here is the question: I have posted a link there to this thread so the OP can view my work.
40. Absolute Extrema: Find f(x,y) over x^2 & y=4

Homework Statement Find absolute extrema of the function over the region R. f(x,y) = 3x2 +2y2 -4y, R: the region in the xy plane bounded by the graphs of y=x^2 and y=4Homework Equations second partial derivative test d=fxxfyy-fxy2The Attempt at a Solution This is my practice test for the final...
41. MHB Find Extrema of f(x,y)=sin(x)sin(y) | Yahoo Answers

Here is the question: I have posed a link there to this thread so the OP can see my work.
42. What Are the Extrema of the Function Based on Its Derivatives?

Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1388504_10201044108366607_730785214_n.jpg?oh=9e67700cd15429886ee87ce2eed63328&oe=528397C9 Homework Equations F(x) = ∫f(x). We can apply the second derivative test. F''(x) = f'(x) The Attempt at a Solution F''(x) is...
43. Absolute Extrema on Open Intervals

Homework Statement Problem 2: Find the absolute extreme values of f(x) on the interval (1, infinity). Homework Equations The Attempt at a Solution It seems as if this is an indiscretion by the teacher. Absolute extrema don't occur on open intervals except in the case of (-∞,∞) such as in...
44. Extrema of Quadratic functions

Homework Statement Does every quadratic function have a relative extrema? Homework Equations Quadratic function: ax^2 + bx + c. Aka a polynomial. Polynomials are continuous through all real numbers. The Attempt at a Solution It seems as if all quadratic functions would have...
45. MHB Find the global extrema of this function over the given region

Hi MHB. Can someone help me with this one please? I've worked out that the critical points are (0,0) and (2,1). But looking at the boundary x = 0, there seems to be no limit to the minimum value. Also, on the boundary y = 1, the value of f(x,1) = -1. So, would I be correct in saying that the...
46. Finding the Nature of Critical Points Using Hessian

Homework Statement Identify and determine the nature of the critical points of the function $$f(x,y,z) = (x^2 + 2y^2 + 1) cos z$$ Homework Equations ##\vec{x}## is a critical point ##\iff Df(\vec{x}) = 0## ##\vec{x}## is a minimum ##\iff## every determinant of upper left submatrix...
47. Local Extrema: Homework Solution Analysis

Homework Statement http://i.minus.com/jZdpOtdOiChOn.jpg Homework Equations Local extrema can be determined using the first derivative test. The Attempt at a Solution I ran the first derivative test to find the critical points, which were 0 and plus/minus 0.5. I plugged in the values into...
48. MHB Emwhy's question at Yahoo Answers regarding finding extrema in a given region

Here is the question: I have posted a link there to this thread so the OP can see my work.
49. Continuous and open function and local extrema

Homework Statement . Let ##f:ℝ→ℝ## be an open and continuous function. Prove that f doesn't have local extrema The attempt at a solution. I suppose there is some ##x_0 \in ℝ## and some ##ε>0## such that ##f(x_0)≤f(x)## for all ##x \in (x_0-ε,x_0+ε)## (the proof for relative maximum is analogue...
50. MHB F's question at Yahoo Answers involving absolute extrema

Here is the question: Here is a link to the question: Math Question - Calculus!? - Yahoo! Answers I have posted a link there so the OP can find my response.