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

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