How do you solve this optimization question?

A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is

y=(kN)/(1+N^2)

where k is a positive constant. What nitrogen level gives the best yield?
N=?

1 Answer
Apr 22, 2018

N=1

Explanation:

Take the first derivative with respect to N:

y'=((1+N^2)k-kN(2N))/(1+N^2)^2

y'=(k+kN^2-2kN^2)/(1+N^2)^2

y'=(k-kN^2)/(1+N^2)^2

Equate to 0 and solve for N:

(k-kN^2)/(1+N^2)^2=0

k(1-N^2)=0

1-N^2=0

N^2=1

N=+-1->N=1 is the only possible answer as we cannot have a negative nitrogen level.

The "best yield" would entail y being at its maximum. To ensure that N=1 gives a maximum for y, evaluate y' in the following intervals:

[0, 1), (1, oo) to determine whether y' is positive (y is increasing) or y' is negative (y is decreasing) in each interval.

If N=1 is a maximum, then y' will be positive before we reach N=1 and negative afterwards:

[0,1):

y'(0)=(k-k(0))/1^2=k>0 So, y is increasing on [0, N)

(1, oo):

y'(2)=((k-4k)/(1+4)^2)=-(3k)/25<0 So, y is decreasing on (1, oo) and the maximum possible crop yield happens with N=1.