Extrema Definition and 153 Threads

Homework Statement Edit: Not absolute, just extrema I've already found the critical point to be (-1/2, -1/4, 1/2) with a value of -1/2. My only problem is finding whether this is a max or min. What technique do I use to find out? I don't believe I can use the 2nd derivative test because all...

Hello guys, Recently I came across a definition to which I'd never given much thought. I was reading through Gelfand and Fomin's "Calculus of variations" and I read the part about weak and strong extrema, and I really can't manage to wrap my head around these definitions. They can be found in...

Homework Statement Find the absolute extrema of the function on the set D. f(x,y)= x^2 + 4y^2 + 3x -1 D= {(x,y) l x^2 + y^2 ≤ 4} Homework Equations The Attempt at a Solution The only thing I've done so far was find the critical point. I found f-sub x = 2x+3 and f-suby= 8y...

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

Hello everybody, I'm new to this forum so thanks for having me. I'm trying to find the times when the extrema occur for a periodic wave f(t) equal to the sum of three sine waves. Given f(t) = sin(2∏at) + sin(2∏bt) +sin(2∏ct) where a, b and c are whole numbers in lowest form (i.e...

Homework Statement f = x^2 + 4xy + y^2 + 6x + 8 Find minimum, maximum, or saddle point Homework Equations A = [f_xx, f_xy; f_yx, f_yy] The Attempt at a Solution Found the determinant to be zero ( i got [2 ,4; 4, 2] ) so i can't use the eigenvalues to determine the extrema. now...

First time poster so I apologize any problems with location or content. By the way, you guys are awesome. Just some context: last semester I had a theoretical mechanics class and the teacher said, while teaching us about the Lagrangian formulation, that particles cannot, ever, reach a...

Homework Statement How may I find stationary points, increase/decrease intervals, concavity for f(x)=(x^3-2x^2+x-2)/(x^2-1)? Homework Equations The Attempt at a Solution I am familiar with how it should be done, except that here I get f'(x)=x^4-4x^2+8x-1 for the numerator of the...

Homework Statement Find the local extrema of the function f(x)=(x^3-2x-2cos(x))Homework Equations derivatives, some algebra The Attempt at a Solution Well, the concept is simple. Solve for the first derivative and set it equal to zero: dy/dx=2sin(x)+3x^2-2=0 and Next, solve for x to...

Homework Statement Locate the absolute extrema of the given function on the indicated interval. f(x) = -x^2 + 3x ; interval: [0 , 3 ] Homework Equations The Attempt at a Solution f'(x) => -2x + 3x = 0 => x = 0 f(0) = -(0)^2 + 3(0) = 0 f(3) = -(3)^2 + 3(3) = 0 My answer: Absolute...

Homework Statement f(x,y) = x^3 - 3x^2 - 6xy + 7y + y^2, x>=0, y>=0 i) Explain why f attains its minimum value on the quadrant. ii) Find the critical points and classify them Homework Equations df/dx = 3x^2 - 6x -6y df/dy = -6x + 7 +2y d^2f/dx^2 = 6x-6 d^2f/dy^2 = 2 d^2f/(dxdy) =...

Hi guys. Today was my last day of Calculus I class, and I’ve had this question simmering in my mind for some time now. Perhaps you guys may enlighten me or point me in the right direction? Consider the Weierstrass function, which is continuous everywhere but differentiable nowhere. We know that...

Homework Statement Find absolute maxima and minima of the function in the given region: T(x,y) = x2 + xy + y2 - 6x Region: Rectangular plate given by: 0 ≤ x ≤ 5, -3 ≤ y ≤ 3 Homework Equations First derivative test, fx =0, fy = 0 Second derivative test, fxxfyy - fxy2 = ? The...

Homework Statement See uploaded file. PROBLEM 5 Homework Equations See uploaded file The Attempt at a Solution I am confused on how to go about doing the rest of this I realize I can do it with the first and second derivatives but I forgot what exactly I do with that. I know the...

How do I find the extrema using Taylor Series?? I am so used to find extrema just by finding the first derivative (make it =0) and then finding the second derivative and then just use the formula f_xx.f_yy - f_xy and just look at the sign but this time I need to use taylor expansion. I hope you...

Hello, I need to find (if there are) minimum and maximum values of the following function: z=\frac{1}{x}+\frac{1}{y} subject to constraint: \frac{1}{{x}^{2}}+\frac{1}{{y}^{2}}=\frac{1}{{a}^{2}} a\neq 0 I think there are no extrema, but I do not know how to show it.

Homework Statement f(x,y)=5xy-7x^2+3x-6y+2 [b]2. Homework Equations (f_xx)(f_yy)-(f_xy)^2 the hessian or discriminant of f The Attempt at a Solution i arrived at a solution but i don't think its correct, and the answer isn't in the back of the book, so i just wanted to ask if i did...

"Show that if a>b>c>0, then the function $f(x,y,z) = (ax^{2} + by^{2} + cz^{2})e^{-x^{2}-y^{2}-z^{2}}$ has two local maxima, one local minima, and four saddle points"

Homework Statement Let h:(0,1)→ℝ be defined by h(x)=cos(\pi\\/x). Determine the set of all x such that x has a relative extreme value at x. Homework Equations The Attempt at a Solution It is common knowledge that the min and max are -1 and 1. Using intuition, it is clear by...

I'm reading a math book and found a couple of proofs I can't do. 1. Given x \in R^n, a \in R, \sum\limits_{i=1}^n{x_i}=na, prove that \sum\limits_{i \in A}\prod\limits_{j = 1}^k {x_{i_j}} \leq \binom{k}{n}a^k where A = \{i \in \{1, 2, ... n\}^k : i_1 < i_2 < ... < i_k\} which essentially...

Homework Statement I really, really don't understand. My work shows one thing, and a simple look at the function shows another. f(x,y) = .25x^2 + 5y^7 + 6y^2 - 3x Find global max and min if they exist. Homework Equations The Attempt at a Solution Took partial derivatives...

Hi, In high school, I was shown an unconventional but quicker way to find max/mins. I'm not sure how common it is but we did it because we learned to curve sketch without calculus first. Take f'(x) =0, and solve for the roots. Construct a number line and place all roots on the number...

Hi again, my calculus exam is on monday so trying hard to understand the things I'm struggling with. I think I'm almost there with part a - well I can solve it through drawing a graph or using a graphing program - but I'd rather solve it without graphs if possible. Part b however is causing me a...

Homework Statement Consider the surface defined by F(x,y,z)=1/2(x+y)^2+(y+z)^2+(x+z)^2=9 (a) Evaluate grad(F) My Solution: grad(F)=(3x+y+2z, x+3y+2z, 2x+2y+4z) (b) Find the highest and lowest points on the surface (i.e. the points where z attains a maximum or minimum). Problem: So I...

Hey guys, I have learned that for a function f defined on an open interval I(a, b), an absolute extremum may not occur. Is this because the domain may take on every value greater than a or less than b without ever equaling a or b? [EDIT] Disregard the question above. I just realized. For a...

Homework Statement Show that the function f(x) = x\sin\left( \frac{\pi}{x} \right) has an infinite number of relative extrema. Homework Equations The Attempt at a Solution Using the product rule and chain rule to find the derivative f'(x): f'(x) = \sin\left( \frac{\pi}{x} \right) -...

Homework Statement show that for any triangle inscribed in a circle with radius r , the equilateral triangle has the largest perimeter. I'm supposed to use the larangian method. Homework Equations The Attempt at a Solution ok so my problem is forming an equation for the perimeter in terms of...

Homework Statement find the local extrema of f(x,y) = 6x^2 -8x + 2y^2 - 5 around the closed disk x^2 +y^2 =< 1Homework Equations The Attempt at a Solution I used the larangian way of doing this but not sure if its right. My solution: F(x,y) = f(x,y) + lambda g(x,y) where g(x,y) = x^2 + y^2...

Homework Statement For what real values of the parameters a,b,c,d does the functiob f(x,y)=ax^3+by^3+cx^4+dy^4-(x+y)^5 have a local minimum at (0,0)Homework Equations I calculated the gradient at (0,0) and it is always zero regardless of parameters. The problem is that the Hessian matrix is...

Homework Statement Find the relative extrema of the following function f(x) = (a-x)/(x2-a2) where a is a constant, a>0 Homework Equations Derivative of f(x), zeroes, quadratic formula The Attempt at a Solution I think I just screwed a small step in there because my answer doesn't...

Homework Statement Sketch the graphs of the following function showing vertical and horizontal asymptotes and relative extrema: f(x) = (x2-1)/(x2-4) Homework Equations Limits, zeroes, derivatives The Attempt at a Solution I know that I have the majority of the answers right, the...

Good day! Can anyone pls teach me how to find the extrema of this function (general second degree polynomial function): f(x,y)=x^2+4xy+4y^2-2x+3y+6. actually, my problem is i don't know how to find the extrema of functions of this form f(x,y)=ax^2+bxy+cy^2+dx+ey+f. Your help will be highly...

How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite. I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no...

Homework Statement i have to find the extrema for (x-1)x^(2/3) so i found the derivative: (5x/3)^2/3 - 2/(3x^1/3) now i have to find the critical numbers by putting the derivative equal to zero and solving for x...but i don't know how to do it. are there even any roots?

Hello, I'm been stuck on this problem and I've been staring blankly at it way too long. I stumbled upon here and thought I'd ask for help? :P Alright well, I'm looking for a local max/min, and I've already done the first partials and I got *f(x)=2x-y and f(y)=-x+2y+6; I'm sure those are right...

Homework Statement A corporation manufactures candles at two locations. The cost of producing x1 units at location 1 is C1= 0.02X12+4X1+500 and the cost of producing x2 units at location 2 is C2=0.05X22+4X2+275. The candles sell for $15 dollars per unit. Find the quantity that should be...

Homework Statement f(x) = x^3/(x^2+ 1) 1. Identify Domain and Range 2. x and y intercepts (if any) 3. local and global extrema 4. Equations of the asymptotes (if any) Homework Equations f(x) = x^3/(x^2+ 1) The Attempt at a Solution f(x) = x^3/(x^2+ 1) f'(x)...

Homework Statement Given function f(x,y) = 3x2 - 2xy + 2y a) find the critical point(s) of f, b) sketch the region R: the triangular region in the xy-plane with vertices (0,0) (0,6) and (3,6) c) find the absolute maximum and absolute minimum of f over R Homework Equations The Attempt at a...

Hello, can i get help with this? Find any relative extrema of the function g(x)=x sech(x) First i find the derivative g'(x)=sech(x)(-xtanh(x)+1) set to 0 sechx=0 can't be solved -xtanh(x)+1=0 xtanh(x)=1 I know i suppose to solve that to get the critical points and then do...

Hello All, My problem is to find the relative extrema of abs(sin 2x); 0< x < 2pi. I got the derivative right, I think: 2 cos 2x if x > 0, and -2 cos 2x if x < 0. However, the solution says that f ' (x) does not exist at x = pi/2, x = pi, and x = (3pi)/2. Glancing at my...

1. find the extremas and points of inflections 2. g(t)= 1+(2+t)e^(-t) 3. So i know you need to find g(t)' and g(t)'' g(t)'= -e^(-t)(1+t) g(t)''= (t)e^(-t) my critical point is t=-1 (max) and my point of inflection is t=o How do I get to max extrema being (-1, 1+e) and...

If it is said that a function has nondegenerate extrema does this simply mean that the extrema are isolated? (The function in question is of one variable.) Thanks in advance

Homework Statement Use a calculator to estimate any extrema of this function: f(x)=5x^3-30x^2+45x+5(sqrt(x)) Homework Equations The Attempt at a Solution I don't know how to find it using a calculator. f'(x)= 15x^2-60x+45+(5/2)x^(-1/2)

Homework Statement Find the relative extrema of each function, if they exist. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. 25. f(x)=1-x2/3 Homework Equations The Attempt at a Solution f prime of (x)= -2/3x-1/3...

given this function f(x) = x^3+y^3+3x^2-18y^2+81y+5 i should i find the extrema(i hope this is how it is called in english) so we should have gradf = 0 hence 3x^2+6x = 0 and 3y^2-36y+81 = 0 here i get x = 0 or x = -2 and y = 54/9 or y = 2 now what i want to ask is, which...

Homework Statement Find the extrema of f subject to the stated constraint: f(x,y) = x-y subject to x2-y2=2Homework Equations Apply the Lagrange Multiplier!The Attempt at a Solution This question was rather odd... I just did a problem similar to this one, and I got the answer right. Let...

Hi, I have a problem solving this question, I don't know if I am on the correct track for solving this. Suppose that the sum of the surfaces of a sphere and a cube is constant. Show that the sum of their volumes is smallest when the diameter of the sphere is equal to the length of an edge of the...

Homework Statement F(x,y)= sin(x)sin(y)sin(x+y) over the square 0< x< pi and 0< y< pi(The values for x and y should be from 0 to pi INCLUSIVE) Homework Equations The Attempt at a Solution partial derivative in terms of x = siny[cosxsin(x+y)+sinxcos(x+y)] you get y=0, pi because siny =0, but...

Homework Statement F(x,y)= sin(x)sin(y)sin(x+y) over the square 0\underline{}<x\underline{}<pi and 0\underline{}<y\underline{}<pi (The values for x and y should be from 0 to pi INCLUSIVE) Homework Equations The Attempt at a Solution I know I need to do the partial derivatives...

Homework Statement If my answers to the questions in the attachments could be checked that would be great. All work is done, and the questions in the pictures are fairly quick and simple. I want to be sure I got them correct. Thank you! Homework Equations The Attempt at a Solution