How do you solve this optimization question?

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=?#

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Answer 1 · 2018-04-22T23:48:05

#N=1#

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#.