Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.

Applied Maximum and Minimum: Calculus?

How should two nonnegative numbers be chosen so that their sum is 1 and the sum of their squares is (a) as large as possible, (b) as small as possible.

3 Answers

Relevance
  • Anonymous
    1 decade ago
    Favorite Answer

    x + y = 1

    y = 1 - x

    g = x^2 + (1 - x)^2 = x^2 + 1 - 2x + x^2 = 2x^2 - 2x + 1

    g' = 4x - 2 ; 0 = 4x -2 ; 2 = 4x ; x = 1/2 and therefore y = 1/2

    so theres a max or min at x = 1/2

    g'' = 4, so it is concave up therefore x = 1/2 is a minimum

    and because there are no maximum, we know that the highest possible value of g are at its boundaries ( end points )( either x = 0 or x =1). In this case x =0 and x = 1 give the same g so they are both the highest possible value of g

  • 1 decade ago

    let the two numbers be x and y

    x^2 + y^2 = a where a is a constant

    x+y=1

    a) x^2 + y^2 =a

    2x + 2y(dy/dx) = 1

    dy/dx = (1-2x)/2y

    when eqn is max or min, dy/dx = 0

    (1-2x)/2y = 0

    1-2x = 0

    x = 0.5

    y = 0.5

    To find the max value, simply look at the graph of the function

    x^2 + y^2 = a

    This is the equation for the graph of a circle with center (0,0)

    when value is max or min, dy/dx = 0, which means the gradient is 0

    The only part of the graph where this happens is on the y-axis ie when x=0 and y = 1 or y=-1. but as your numbers must be positive, y=1

  • Anonymous
    1 decade ago

    Let the two numbers be a and b

    then a+b=1

    b=1-a

    Sume of their squares - S=a^2+b^2

    subs in for b

    S=a^2+(1-a)^2

    S=2a^2-2a+1

    dS/da=4a-2

    for max and min put dS/da=0

    4a-2=0

    a=1/2, b=1/2 these values give minimum

    maximum is when one is 1 and other is 0.

    If you plot the graph f(a)=2a^2-2a+1 you will see that the maximum values are not turning points so will not be calculated by this method.

Still have questions? Get your answers by asking now.