find two numbers whose sum is 30 and whose product is a maximum?

Calc 1 way:

f = xy
If x+y = 30, then
f = x(30-x)
f = 30x - x^2
f' = 30 - 2x
f''<0
0 = 30 - 2x
2x = 30
x = 15
y=15

lagrange multipliers way:

f(x,y) = xy
g(x,y) = x+y

∇f = λ∇g
<y,x> = λ<1,1>

λ = x = y. (substitute into constraint)

2x = 30
x = 15
y=15...Show more

x + y = 30
=> y = 30 − x

P = xy
=> P = x(30 − x)
=> P = 30x − x^2
=> dP/dx = 30 − 2x
=> d^2P/dx = -2

P is at maximum when dP/dx = 0 and d^2P/dx^2 < 0

0 = 30 − 2x
=> 2x = 30
=> x = 15
=> y = 30 − 15
=> y = 15

x = 15, y = 15...Show more