Extrema Definition and 151 Threads

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 $x_1<x_2<x_3<x_4$...

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...

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...

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...

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...

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?

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...

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 $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...

$\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...

$\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*}...

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...

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

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...

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*}...

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...

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?

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...

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...

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...

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...

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...

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 =...

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...

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...

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...

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?

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...

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...

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...

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?

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...

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...

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...

Here is the question: I have posted a link there to this thread so the OP can view my work.

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...

Here is the question: I have posted a link there to this thread so the OP can view my work.

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...

Here is the question: I have posted a link there to this thread so the OP can view my work.

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...

Here is the question: I have posed a link there to this thread so the OP can see my work.

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...

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...

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...

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...

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...

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...

Here is the question: I have posted a link there to this thread so the OP can see my work.

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...

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.

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...