Maximum of a product
When is the product at its maximum, if and are related by , where is a fixed positive constant?
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1 solution
x + 2 y = a
y = ( a − x ) / 2
Plugging in:
f = x y = ( x a ) / 2 − x 2 / 2
To find the extreme points of this function, we can determine when the rate of change (derivative) is zero:
f ′ ( x ) = a / 2 − x = 0
x = a / 2
By observing that at the endpoints x=0 and x=a the product x*y is zero, we can conclude that x=a/2 is a maximum. (One could have also used the 2nd derivative here)