What is Maxima and minima: Definition and 49 Discussions

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire domain (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

View More On Wikipedia.org
  1. WMDhamnekar

    MHB Maxima and Minima (vector calculus)

    Hi, Hi, Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$ But when I read this graph, I couldn't guess that the stated function have a...
  2. WMDhamnekar

    MHB Maxima and Minima in calculus

    Question: Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone. Answer: Let r and h be the radius and height of the right circular cylinder inscribed in a given cone of radius R and height H. Let...
  3. A

    Finding the Local Max/Min of f(x,y) on C

    Hi everyone, I'm struggling with this problem: Let ##f(x,y) = \begin{cases} (x-y)\ln(y-x) & \text{if } y>x \\ 0 & \text{if } y\leq x \end{cases}## and let ##C=\{(x,y)\in \mathbb{R}^2|x^2+y^2=1\}## Then proof that ##max_Cf=1/e## and ##min_Cf=-(\ln2)/\sqrt2## My solution: I used Lagrange...
  4. T

    Maximizing Transferred Energy between n bodies in linear collisions

    Homework Statement My problem has two parts. 1) We have two point masses ##m,M##. and there is another mass ##m_1## between them.They are all aligned in a line. Mass ##M## is moving with speed ##u_1## toward ##m_1## and after collision and all other masses are not moving. we want to find...
  5. E

    Verify that the sum of three quantities x, y, z

    Homework Statement Verify that the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal. Homework Equations w = x + y + z k = x * y * z The Attempt at a Solution Assuming that my understanding of the question is correct i.e. that we...
  6. Matejxx1

    Maxima and minima and finding the radius of the circle

    Homework Statement Find where θ is the biggest (largest) I'll have the picture of the problem included below (pic:1) Homework Equations (x-q)2+(y+5/2)2=r2 answer x= 2 The Attempt at a Solution Hi, so my prefesor gave me this problem and told me to try to solve it. We already did this problem...
  7. L

    MHB Maxima and minima of a triangle

    Write the following formulas: a) The minimum perimeter of any triangle (abc) only known heights corresponding to the sides a and b. b) The maximum height and minimum corresponding to the side b of any triangle (abc) only known the value of its perimeter and height corresponding to the side a.
  8. B

    MHB Finding the maxima and minima of a function

    I have the following function: f(x,y) = xye(-x2-y2) I am trying to find all the maximum and minimum points. I have started off by finding the partial derivatives of the function and equation them to zero. df/dx = ye(-x2-y2)(1-2x2) = 0 df/dy = xe(-x2-y2)(1-2y2) = 0 However, i am stuck for...
  9. P

    Maxima and Minima, Rod Rotating on a Surface With Friction

    Homework Statement A uniform rod of mass M and length L can rotate around point P which is at position x from one end of the rod as shown in the figure. The rod is gently placed on a rough horizontal surface that has a friction coefficient μ. and at t=0 starts rotating with angular velocity...
  10. K

    Classifying Critical Points: Finding Local Extrema and Saddle Points

    Homework Statement Homework EquationsThe Attempt at a Solution 1) I found the asymptote as (+/- 1) 2) Let f(x) = y; dy/dx = -2x^2 / (x^4 - 2x^2 + 1) = 0 -2x^2 - 0 x = 0; Since f() != 1, f(2) > 0 Increasing Since f() != -1, f(-2) < 0 Decreasing So i guess range is increasing or x >=2...
  11. trash

    Maxima and minima of differential equation

    Homework Statement Consider the differential equation y'=x-y^2. Find maxima, minima and critical points; show that for every solution f=f(t) exists T\geq 0 such that f(t)< \sqrt{T}\;\forall t > T Homework Equations The Riccati equation: y'=a(x)y^2+b(x)y+c(x) The Bernoulli equation...
  12. C

    Proving the Local Maxima of a Rectangle Partition with Two Parallel Lines

    A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles. Unless I did incorrectly, the algebra is very very long... HELP
  13. S

    Maxima and Minima of a function of several variables

    Homework Statement Find the maxima and minima of: f(x,y)=(1/2)*x^2 + g(y) g∈⊂ (δ⊂ ℝ ) in this region Ω={(x,y)∈ℝ2 / (1/2)*x^2 + y^2 ≤ 1 } hint: g: δ⊆ ℝ→ℝ The absolute min of f in Ω is 0 The absolute max of f in Ω is 1 Homework Equations The Attempt at a Solution I have the...
  14. C

    Maxima and Minima of two-variable functions word problem

    1. Problem: A manufacturer makes two models of an item, standard and deluxe. It costs $40 to manufacture the standard model, and $60 for the deluxe. A market research firm estimates that if the standard model is priced at x dollars, and the deluxe at y dollars, then the manufacturer will sell...
  15. L

    Optimizing Distance: Finding the Minimum Value and Coordinates on a Line

    A point P whose x-coordinates is a is taken on the line y=3x-7. If Q is the point(4,1) show that PQ2 = 10a2-56a+80. Find the value of a which will make the expression a minimum. Hence show that the coordinates of N, the foot of the perpendicular from Q to the line are (24/5 , 1 2/5). Find...
  16. J

    Maxima and minima word problem

    Homework Statement A corridor of with a is at right angles to a second corridor of width b. A long, thin, heavy rod is to be pushed along the floor from the first corridor into the second. What is the length of the longest rod that can get around the corner? Homework Equations I...
  17. M

    Maxima and minima of functions of two variables

    Homework Statement Locate all relative maxima, relative minima ,and saddle points if any. f(x,y)=ysinx fx(x,y)=ycosx fy(x,y)=sinx ycosx=0 sinx=0 y=0 x=0,∏,2∏... up until infinity Critical points at (0,0),(∏,0),(2∏,0)... fxx(x,y)=-ysinx...
  18. J

    What Is the Correct Derivative of the Function y(r) = (r^2 - 5.50r)exp(-r)?

    Homework Statement Consider the following function of the variable r, r>/=0 y(r)=(r^(2)-5.50r)exp(-r) Find the value of the first derivative dy/dr at r=5.50 Homework Equations How do I solve this? I know its a simple derivative equation, but I can't seem to get it. I tried finding the...
  19. S

    Finding the maxima and minima of a surface defined by datapoints

    Here's the problem I'm trying to solve, We have an N dimensional surface. We do not know the form of this surface however we have datapoints which are likely to be close, though not guaranteed to be on the surface, with some outliers. What I want to do is determine where in N dimensional...
  20. R

    Is there a way to prove that A(x) is minimum when x=2r?

    The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. is this way correct? assume that x is side of square and r is radius of circle k=4x+2∏r sum of...
  21. D

    Calculus 1: Finding the maxima and minima of the function 1-x^(2/3)

    Homework Statement I am given the problem "A function is defined by f(x) = 1-x^(2/3). a. find the minima and maxima of the function. State whether they are relative or absolute. b. graph the function." Homework Equations I found the derivative and set it equal to zero. y' =...
  22. J

    What are the local maxima and minima of F(x)=(x^2)/(x+1)?

    Homework Statement F(x)=(x^2)/(x+1) Find critical points Find local maxima & minima Homework Equations None The Attempt at a Solution F'(x) = x(x+2)/(x+1)^2 crit points: -2,0,-1 f(-2) = -4 f(0) = 0 f(-1)=undef My book is telling me that f(0) is the minima, and f(-2) is the...
  23. K

    Find All local maxima and minima and all saddle points of the function

    Homework Statement ## f\left( x,y\right) =x^{2}-4xy+6x-8y+2y^{2}+10 ## ## f_{x}=2x-4y+6=0 ## ## f_{y}=-4x-8+4y=0 ## ## f_{y}=-4\left( x-y+2\right) ## -2=x-y, then solving fx and using this equality ## f_{x}=0=2\left( x-2y+3\right) =0 ## 2(-2 -y+3)=0 2(1-y)=0 y=1, then pluggin...
  24. Telemachus

    Proof of Saddle Roof at (0,0) for f(x,y)=(y-3x^2)(y-x^2)

    Hi there. I've got this function f(x,y)=(y-3x^2)(y-x^2), and I have to analyze what happens at (0,0) in terms of maxims and minims. But what I actually have to proof is that there's a saddle roof at that point. Theres is a critical point at (0,0). Let's see...
  25. A

    Michelson interferometer - maxima and minima

    URGENT! michelson interferometer - maxima and minima Homework Statement If a wave is reflected at a surface of a plate with higher refractive index (eg. At air glass or air-metal) it suffers a λ/2 phase change. Show that for the interferometer, maxima will occur for 2dcosθ = (n+1/2)λ and...
  26. Telemachus

    Hessian matrix, maxima and minima criteria

    Homework Statement Hi there. I've got some doubts about the maxima and minima on this function: f(x,y)=x \sin y. I've looked for critical points, and there's only one at (0,0). The thing is that when I've evaluate the second derivatives I've found that f_{xx}=0, then I have not a defined...
  27. Z

    Maxima and minima properties problem

    Homework Statement Let An be the nth term of an A.P. and if A7 = 15, then the value of the c.d. that would make A3 x A7 x A12 greatest is : 1)9 2)9/4 3)3/8 4)18 Homework Equations The Attempt at a Solution Applying AM>=GM A3+A7+A12/3 >= (A3 x A7 x A12)^1/3 given that...
  28. V

    Maxima and Minima application question

    Homework Statement A farmer is planning to build six adjoining rectangular pens of equal size to house his hens, as shown in the diagram. He only has 180m of fencing, however, and he wants to make the pens as large as possible. Find the maximum area he could make each pen. The diagram is...
  29. N

    Absolute vs. Relative Maxima and Minima

    Homework Statement I do not have a specific homework problem, but could someone please clarify this for me? QUESTION: When you have an absolute maxima (or minima), how can you tell if it is ALSO a relative maxima (or minima)? I understand how to find absolute extrema on a closed...
  30. T

    Maxima and Minima on surfaces in three dimensional space

    Homework Statement Find the maximum and minimum values of f(x,y) = (xy)2 on the domain x2 + y2 < 2. Be sure to indicate which is which Homework Equations I am not sure what to put here. I solved this problem a different way, and I am not confident I did it correctly. The Attempt at a...
  31. K

    Calculators Finding Maxima and Minima with TI-89

    I am new, but i have a question about the TI -89 Titanium calculator, does anyone how if i can plug in the equation f(x)=2x^3-15x^2+36x-1 to identify each location of the local maximun and minimum, and inflexion poits? Will this calculator do this?
  32. T

    Find Absolute Maxima and Minima of f(x) on [-2,2]

    1. Find the absolute maxima and the absolute minima of the following function f(x)=(2x)/(x^2+1) on [-2,2]
  33. 3

    How Do You Determine Lines of Maxima and Minima Based on Gradient?

    Homework Statement Please look at the attached picture In that I had to draw the line of maxima and minima Homework Equations line of maxima has greatest gradient line of minima has smallest gradient The Attempt at a Solution which line should be the maxima? the line with the...
  34. S

    Maxima and Minima of z=f(x,y) theorems

    Hi all, In my Calculus III course, we are using Stewart's book, so as you might know there is not much rigor in there. Likewise, when it came to the section on Maximum and Minimum values of a function with two variables z=f(x,y), they ommited a lot of stuff. Hence, i tried to use...
  35. E

    Optimization with maxima and minima

    A rectangle is to be inscribed in a right triangle having sides 6 inches, 8 inches, and 10 inches. Determine the dimensions of the rectangle with greatest area. I recently tried doing it and the answer was found by finding the slope and then using the first and second derivatives of the area...
  36. N

    Maxima and Minima of a function

    Are there any analytical techniques to do this besides the Derivative Test?
  37. J

    How Do You Calculate the Profit-Maximizing Price for a Monopolist?

    Homework Statement For a monopolist's product, the demand equation is: p = 42 - 4q and the average-cost function is c = 2 + (80/q) Find the profit-maximizing price. Homework Equations When i started to solve the problem, i deduced...
  38. C

    How Do Slit Separation and Width Affect Diffraction Patterns?

    Does the distance between the maxima increase/decrease or stay the same when the slit separation is increased? when the slit width is increased? what about if it was minima? thanks for any help to understand how this works.
  39. J

    Applied Maxima and Minima Problem

    Homework Statement Problem goes like this: For a monopolist's product, the demand equation is: p=156-2q _ and the average-cost is c(ave)=120+112/q Homework Equations We need to find the total cost function in terms of c= The Attempt at a Solution...
  40. I

    Find the maxima and minima of the magnitude

    Homework Statement A hypothetical star S has a planet P that moves in an elliptical orbit with period T . The minimum distance of P from S is d and the maximum distance is 1.7d. (i) find the maxima and minima of the magnitude of the acceleration of P, in terms of d and T . Homework...
  41. R

    Maxima & Minima: Inscribing Cylinder in Sphere of Radius R

    How would we find the maximum surface area of a cylinder inscribed in a sphere of radius R. This problem is given in my textbook . I know concept of maxima and minima will be apllicable here but i can,t start and make the expression of surface area in a suitable manner. Anybody having answers
  42. G

    Solve Shortest Distance Point (0,c) from Parabola y=x^2

    Hi I am stuck on this ques. although i have solved and got the ans but i am still facing a problem. Find the shortest distance of the point (0,c) from the parabola y=x^2 where 0<= c <= 5 i got the ans right that is (c-0.25)^0.5 but i have not been able to understand the use of the 0<= c <= 5...
  43. O

    Maxima and Minima for a two variable function

    Greetings, can you guys please help me with my assignment?? I am supposed to find and classify all critical points of the function f(x,y) = (sin x)(cos y) Now I took the first partials with repsect to x and y and they are (cos x)(cos y) = 0 and (-sin x)(sin y) = 0, respectively. Now I...
  44. C

    How Do You Optimize Box Volume by Cutting Squares from a Sheet of Metal?

    Hello all You are planning to make an open box from a 30- by 42 inch piece of sheet metal by cutting congruent squares from the corners and folding up the sides. You want the box to have the largest possible volume. (a) What size square should you cut from each corner? (gice side length of...
  45. C

    How to find increasing and decreasing intervals of a biquadratic function?

    Hello all Given y = -x^4 + 18x^2 + 11 how would you find the intervals where the function is increasing and decreasing? Would you have to find f'(x) and find critical points? Thanks
  46. C

    Applied Maxima and Minima Problems

    Hello all I have a few questions on applied maxima and minima 1. A company gives you 675 sq. ft of cardboard to construct a rectangular carton with the largest volume. If the carton is to have a square base and an open top, what dimensions would you use? My Thought Process: Volume =...
  47. H

    A confusing prob. on maxima and minima

    I have u(x,t)=-2xt-x^2 find maximum in region {-2 ≤ x ≤ 2 , 0 ≤ t ≤ 1} I believe to find the critical point first I have to take the partial derivative with respect to x and t and equate to zero. Thus Ux=-2t-2x = 0 Ut=-2x = 0 Thus the only critcal point I find is x=0, t=0. But the...
  48. S

    Calculus maxima and minima word question can't understand

    Calculus maxima and minima word question...can't understand :( A rectangular field is going to be enclosed and divided into two separate rectangular areas (not equal either). Find the minimum fencing required if the total area of the field is 1200m^2 (See the picture attached right now)...