Answer:
Step-by-step explanation:
Given that three positive numbers have sum 18.
Let the numbers be [tex]x,y, 18-x-y[/tex]
Then product
[tex]f(x,y) =xy(18-x-y) = 18xy-x^2y-xy^2[/tex]
To find maxima, let us use partial derivaties
[tex]f_x = 18y-2xy-y^2\\f_y =18x-x^2-2xy\\f_xx= -2x\\f_yy=-2y\\f_xy = 18-2x = f_yx[/tex]
Equate I derivatives to 0
Solving the two linear equations we get solution as
(6,6)
Hence maximum when x =y=z=6
i.e. when all numbers are equal to 6.
