The sum of two numbers is 64. Show that their product has a maximum of 1024.

Question

please show detailed working out I have trouble understanding this even though it looks easy.

Answer 1

x+y =64

y=64-x

xy= x(64-x) = 64x -x^2

take derivative and set = 0 to maximize xy

64-2x = 0

2x =64

x =32

y=64-32 = 32

max xy = 32(32) = 1024

this may be more of a calculus than algebra problem

but more algebraically, with no calculus

x+y = 64

xy = 1024

x=1024/y

1024/y + y = 64

1024 + y^2 =64y

y^2 -64y + 1024 = 0

y = 64/2 + or - (1/2)sqr(64^2-4(1024)) =

32 + or - (1/2)sqr(4096-4096) = 32

y=32

x =32

xy = 32^2 = 1024

move away from x=y slightly and the product is less

31(33) = 1023

go further from equality and the product approaches zero

63(1) = 63

63.5(1/2) = 31.75

63.99(.01) = 0.6399

63.9999(.0001) = 0.00639999