What is Extrema: Definition and 153 Discussions

Homework Statement Find the extrema of the function subject to the given constraint. f (x, y) = x x2 + 2y2 = 3 Homework Equations det = fxx * fyy - (fxy)2 If det > 0, fxx < 0 \Rightarrow MAXIMUM fxx > 0 \Rightarrow MINIMUM If det < 0, \Rightarrow SADDLE POINT If...

Homework Statement Find any extrema, points of inflection, asymptotes, and symmetry for function. Homework Equations f(x) = (x^5-10x^3+9x) / ( x^4 - 16) The Attempt at a Solution Extrema: I took the first derivative by using the Quotient Rule, and got (x^8 + 10x^6 - 107x^4 +...

Homework Statement Considering the surface of a sphere of radius 1 with its centre at coordinates (0,0,0). For the function: f(x,y,z) = x^{3} + y^{3} + z^{3} Need to find the following: (i) All the extrema on the surface which have x, y and z all non-zero simultaneously...

Homework Statement I have to find a, b, c, and d such that the function defined by : f(x) = ax3+bx2+cx+d will have a relative extrema at points (1,2) and (2,3). The Attempt at a Solution From the given critical points, I am able to know that when x=1 or x=2, f'(x)=3ax2+2bx+c should be...

Please check my steps for global extrema of functions. 1) Finding Critical points. 2) Finding values of f at critical points and end-points. 3) Checking which one is largest and which is smallest. 4) Largest would be Global maxima and the smallest would be minima. Now a question, i know...

how can i find the x-coordinates of all local extrema in this equations please anyone could answer it for me? f(x)=x^3+4x^2+2x f(x)=x^4-3x^2+2x f(x)=x^5-2x^2-4x f(x)=x^5+4x^2-4x

how can i find the local extrema of x-coordinates in this equation? f(x)=x^4-3x^2+2x

Use any method to find the relative extrema of |1+\sqrt[3]{x}|. I get the following |1+\sqrt[3]{x}| = \left\{\begin{array}{cc} 1+\sqrt[3]{x}, & \mbox{if} x >-1 \\ -1-\sqrt[3]{x}, & \mbox{if} x<-1\end{array}\right Using the first derivative test, I get f'(x) = \frac{1}{3x^{2/3}} \mbox{and}...

Find the relative extrema of f(x)=|3x-x^2|. Normally you solve f'(x)=0 to find the critical points, in that case we have 3-2x=0 => x=3/2. However there are two other relative extrema that I cannot find by calculus as the absolute nature of f(x) has confused me. Indeed doing a sign analysis on...

Quick question: Suppose I hold two initially synchronized clocks on Earth and throw one up and catch it when it comes back down. Now my (small amount of) knowledge of GR tells me that the proper time on the thrown clock should be maximized since it was on a geodesic. However, this seems like...

Homework Statement When you cough, your windpipe contracts. The speed, v, with which air comes out depends on the radius, r, of your windpipe. If R is the normal (rest) radius of your windpipe, then for r <_ R, the speed if given by v = a(R-r)r2 where a is a positive constant. What value of r...

Homework Statement f(x)={\frac {{x}^{2}-2}{{x}^{2}-1}} List the critical points, extrema, points of inflection, and concavity. Homework Equations f'(x)={\frac {2x}{{x}^{2}-1}}-{\frac { \left( 2\,{x}^{2}-4 \right) x}{ \left( {x}^{2}-1 \right) ^{2}}}...

Homework Statement Find and classify the extrema of f = x3 - 3xy2 + y3 The Attempt at a Solution I find partial deriv. with respect to x is 3x2 - 3y2 and the partial deriv. with respect to y is -6xy + 3y2 I set these to zero, and for my critical point I get (0,0). This can't be...

]I have to find the local/absolute extrema for the following function: f(x,y)=x^2+y^2 bounded by the triangle x=0,y=0,y+2x=2 So far i have: fx(x,y)=2x fy(x,y)=2y fxx(x,y)=2 fyy(x,y)=2 fxy&fyx(x,y)=0 critical pts at (0,0,0) domain: 0<=x<=1, 0<=y<=2 i don't know what i should do...

find this problem really though. I did solve for it's derivative and then by EVT tried to find its crititcal points. I get (-π/2) and (π/2) for f'(x)=0. The problm is that I'm not sure if there are more solutions for f'(x). Please see function in problem 1 of attachment: Any help will...

Homework Statement The cost of producing x units is given by: f(x)=800+110x-110ln(x) Find the minimum average cost. x >= 1 Homework Equations Cost=f(x)/x Marginal cost is C'(x) The Attempt at a Solution This is what I have done so far: C(x)=(800+110x-110ln(x))/x...

Homework Statement Two curves y_1=(\frac{20}{x^2})\sin(\frac{10}{x}) and y_2=5\cos x intersect in three points in the interval (1,3). Draw the graphs, compute the minimum and maximum points as well as the turning points and show the points of intersection. 2. The attempt at a...

F(x)=x^m * (1-x)^n where m and n is greater or equal to 2? find the x coordinates of local extrema? would my answer be with respect to m and n? or do i plug in a number for these variables

hey guys, hope you can help me with this, here's the picture: 1. to find whether it is increasing or decreasing, f'(x)<0 is decreasing and f'(x)>0 is increasing. so first-> i took the derivative and got f'(x)=12x^2+60x-168 and set it to zero to find the critical numbers. I got x= -7 and x=...

Homework Statement Find the global extrema of the function f(x,y)=sinxy on the closed region given by 0<x<pi and 0<y<pi. You must provide a complete analysis of the function both inside the region and on each of the boundary lines. Be sure to clearly indicate the maximum and minimum values and...

[SOLVED] Determining Local Extrema with Lagrange Homework Statement Find local extram of f(x,y,z) = 8x+4y-z with constraint g(x,y,z) = x^2 + y^2 + z^2 = 9 Homework Equations \nabla f(x,y,z) = \lambda g(x,y,z) The Attempt at a Solution So I did the partial derivatives for F and...

Homework Statement a) Let U be a closed subset of the reals with an upper bound. You know that U has a supremum, say z. Prove that z is an element of U. b) Suppose U is a closed subset of the real numbers with an upper and lower bound. Prove that U has a maximum and minimum.The Attempt at a...

I know this is supposed to go in the HW forum but its not working there so I'm trying it here, and I'm actually running into the same problem with another problem AGAIN. Someone tell me if I am doing this right: --- \nablaHomework Statement Find the extrema of f(x,y)=x-y ; subject to...

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is...

Hello, my name is Calleigh and i am new to the forum! I am in Calculus II and have a few questions on some problems. I am using the textbook Calculus 8th edition by Larson, Hostetler and Edwards. Could someone please help me? The problem is on pg 959 in chapter 13.8 in the text, number 56. It...

Homework Statement Show that among all parallelograms with perimeter l, a square with sides of length l/4 has maximum area. Do this using the second partials test, and then using Lagrange multipliers. Homework Equations Area of a parallelogram: A = absin\phi, where a and b are the...

tough one find all critical points of f(x, y)=e^x(1-cos y) and classify these critical points.

1) f(x,y,z)=x3-3x-y3+9y+z2 Find and classify all critical points. I am confused about the following: The Hessian matrix is diagonal with diagonal entries 6x, -6y, 2. Now, the diagonal entries of a diagonal matrix are the eigenvalues of the matrix. (this has to be true, it is already...

{} these brackets are going to represent the absolute value lines the problem states find the absolute extrema of the given function on each individual interval: f(x)= {2x} - {x-2} a) [0,1] b) [-3, 4] I know I need the derivative of the equation but it does not really give a good...

Homework Statement Classify the local extrema of f(x) = x/2 + cos(x), 0 <= x <= 2pi. Give the exact values of the critical numbers and extrema. Find the absolute (or global) maximum and minimum values of the function. The Attempt at a Solution So i need to find the critical numbers...

If we have a function f(x,y) for which we wish to find the stationary points, then there is one set of rules for classifying those points. Let there be an extremum of f(x,y) at (x,y) = (a,b), and D = fxx(a,b)*fyy(a,b) - {fxy(a,b)}², then 1) D > 0 and fxx > 0 => min 2) D > 0 and fxx < 0 =>...

Homework Statement Find the absolute maximum and minimum values of f(x) = e^{}-x - e^{}-2x on [0,1]. f '(x) = -e^-x + 2e^-2x Homework Equations The Attempt at a Solution f '(x) = -e^-x + 2e^-2x 0 = -e^-x + 2e^-2x e^-x = 2e^-2x ln(e^-x) = ln(2e^-2x) -x = ?

Homework Statement Sketch a graph of a function f that is continuous on [1,5] and has no local maximum and minimum, but 2 and 4 are critical numbers. Homework Equations The Attempt at a Solution Knowing 2 and 4 are critical numbers, I formed the base function x^{}2-6x+8. Not sure...

Homework Statement These are two problems from my assignment that are due tomorrow! Plz, help as you can. 1)Show that f(x,y) = (y - x)(y - 3x^2) attains a local minimum on every straight line through the origin and that this occurs at (0,0). Does f have a local minimum at (0,0)...

Homework Statement Find and classify the local and absolute extrema of the function f(x,y)=x-xy over the region D={(x,y)|x^2+y^2\leq1 and x+y\geq0} Homework Equations The Attempt at a Solution Critical points are where the first derivative (gradient) is 0. Grad(f)=(1-y, -x)=0 So critical...

Homework Statement Find and classify the local and absolute extrema of the function f(x,y)=x-xy over the region D={(x,y)|x^2+y^2\leq1 and x+y\geq0} Homework Equations The Attempt at a Solution Critical points are where the first derivative (gradient) is 0. \nablaf=(1-y, -x)=0...

Hi I'm studying for a calculus exam and I'm a little stuck on finding the extrema for multivariable functions. For the particular question I'm trying to do now I need to find and classify the extrema for the function f(x,y) = (4x^2)(e^y) - 2x^4 - e^4y. I can find the first derivatives...

Homework Statement Find the absolute max and min of h(x) = cos(2x) - 2sinx in the closed interval [(pie)/2, 2pie] The Attempt at a Solution I got h(x) = cos(2x) - 2sinx h'(x) = -sin(2x)*2 - 2cosx <---chain rule i did :smile: 0=-2sin(2x) - 2cosx 0=[-2sin(2x)/-2] -...

Extrema Problem - Need help Please help me with this problem if you can: The function f(x) is defined as f(x)=-2(x+2)(x-1)^2 on the open interval (-3,3). 1. Determine the coordinates of the relative extrema of f(x) in the open interval (-3,3) 2. Let g(x) be defined as g(x)= absolute...

Hi, I have a function f(x) = \frac{\cos(x^2)+x}{x^2+2},\quad x\in[0,3] I have to find the extreme values of the function in in the range [0,3], with Maple, by solving f'(x) = 0. Maple will solve these numerical, and I get 3 values. c_1 = 0.5345058769, c_2=1.732313261 and c_3=2.461303848 ...

I need to find the extrema of f(x,y,z)=x+y+z subject to the restraints of x^2 - y^2 = 1 and 2x+z = 1. So the gradient of f equals (1,1,1) = lambda1(2x,-2y,0) + lambda2(2,0,1). Solving for the lambdas I found that lambda1 = -1/(2x) = -1/(2y), or x=y. But this isn't possible if x^2 - y^2 = 1...

For f(x,y) = x^2 + y^2 + 3xy I need to find the critical points and prove whether or not they are local minima, maxima or saddle points. I thought the only critical point was (0,0) since Df = (2x + 3y, 2y + 3x) = 0. Doesn't this make (0,0) a local min? The reason I doubt this now is because upon...

A cylindrical can is to hold 20 m3. The material for the top and bottom costs €10/m^2 and material for the side costs €8/m^2. Find the radius r and height h of the most economical can. the answer is h= 5 and j=2 i have found the eqn such as v=pi j^2 h =20 pi and 2pi j^2 + pi j h =...

http://img33.imageshack.us/img33/9021/graph3jc.jpg Ok, I had a function and the directions were: Locate and classify all extrema as absolute or relative... I can't remember the equation, but when i found the critical numbers of the function.. i got a critical number at the origin.. but...

Hi, I want to find global extremes of function f(x,y,z) = xy^{2}y^3 on the set M = \left\{[x,y,z] \in \mathbb{R}^{3}, x+2y+4z = a, x,y,z > 0\right\} I need to show that this is compact. Because I'm in \mathbb{R}^{n} it is sufficient to show it is closed and bounded. Closeness (is this...

Your firm offers to deliver 300 tables to a dealer, at $90 per table, and to reduce the price per table on the entire order by 25cent for each additional table over 300. Find the dollar total involved in the largest possible transaction between the manufacturer and the dealer; then find the...

The extreme value theorem says that if a function is continuous on a closed interval then there are an absolute max and min on that interval. This means that the function can only be defined on this interval because if it existed outside this interval then the absolute extrema on the interval...

we're doing partial derivatives, and i thought i understood them until: f(x,y)=-x^2-y^2-10xy+4y-4x+2 we are meant to find and classify all local extrema. i got: f_x=-2x-10y-4 and f_y=-2y-10x+4 i've never had two variables at this stage, and i can't solve for any of the points. the...

Hello! I am doing a question about curve sketching and determining extrema and just need some clarification. Here is the question: Determine the extreme values of the function f(x) = x^2 - 8x + 9, where -1 is greater than or equal to x is less than or equal to 5. This is what I have...

I'm supposed to find the absolute max and min for f(x,y)=sin(x) +cos(y) on the rectangle defined by 0<=x<=2*pi 0<=y<=2*pi. Can someone point me in the right direction. I got the critical points but i don't know what to do after that.