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=kN1+N2 where k is a positive constant. What nitrogen level gives the best yield?

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kwabenzafvf · Accepted Answer

Take the first derivative with respect to N:y′=(1+N2)k−kN(2N)(1+N2)2y′=k+kN2−2kN2(1+N2)2y′=k−kN2(1+N2)2Equate to 0 and find N by solving: k−kN2(1+N2)2=0k(1−N2)=01−N2=0N2=1N=±1→N=1 is the only viable response because a nitrogen level that is negative is not possible.The &quot;best yield&quot; would be y at its highest value. Ascertain that y has a maximum when N=1 by evaluating y&#039; over the following intervals:[0,1),(1,∞) to determine whether  y&#039; is positive (y is increasing) or y&#039; is negative (y is decreasing) in each interval.If N=1 is a maximum, then y&#039; will be positive before we reach N=1 and negative afterwards:[0,1):    y′(0)=k−k(0)12=k&amp;gt;0 Therefore, y is increasing on [0,N)(1,∞):y′(2)=(k−4k(1+4)2)=−3k25&amp;lt;0 Then, y is decreasing on (1,∞) and the maximum possible crop yield happens with N=1.

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?

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