Answer:
A nitrogen level of 1 gives the best yield.
Step-by-step explanation:
We are given the following:
Yield Y of an agricultural crop as a function of the nitrogen level N in the soil
![Y(N) = \displaystyle\frac{kN}{1+N^2}](https://tex.z-dn.net/?f=Y%28N%29%20%3D%20%5Cdisplaystyle%5Cfrac%7BkN%7D%7B1%2BN%5E2%7D)
First, we differentiate Y(N) with respect to N, to get,
![\displaystyle\frac{d(Y(N))}{dN} = \displaystyle\frac{d}{dN}\Bigg(\displaystyle\frac{kN}{1+N^2}\Bigg)\\\\= \frac{(1+N^2)k-kN(2N)}{(1+N^2)^2}\\\\=\frac{k-kN^2}{(1+N^2)^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%28Y%28N%29%29%7D%7BdN%7D%20%3D%20%5Cdisplaystyle%5Cfrac%7Bd%7D%7BdN%7D%5CBigg%28%5Cdisplaystyle%5Cfrac%7BkN%7D%7B1%2BN%5E2%7D%5CBigg%29%5C%5C%5C%5C%3D%20%5Cfrac%7B%281%2BN%5E2%29k-kN%282N%29%7D%7B%281%2BN%5E2%29%5E2%7D%5C%5C%5C%5C%3D%5Cfrac%7Bk-kN%5E2%7D%7B%281%2BN%5E2%29%5E2%7D)
Equating the first derivative to zero, we get,
![\displaystyle\frac{d(Y(N))}{dN} = 0\\\\\Rightarrow \frac{k-kN^2}{(1+N^2)^2} = 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%28Y%28N%29%29%7D%7BdN%7D%20%3D%200%5C%5C%5C%5C%5CRightarrow%20%5Cfrac%7Bk-kN%5E2%7D%7B%281%2BN%5E2%29%5E2%7D%20%3D%200)
Solving, we get,
![\displaystyle\frac{k-kN^2}{(1+N^2)^2} = 0\\\\k-kN^2 = 0\\k(1-N^2) = 0\\k \neq 0\\1-N^2 = 0\\N = \pm 1\\N \neq -1\\N = 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bk-kN%5E2%7D%7B%281%2BN%5E2%29%5E2%7D%20%3D%200%5C%5C%5C%5Ck-kN%5E2%20%3D%200%5C%5Ck%281-N%5E2%29%20%3D%200%5C%5Ck%20%5Cneq%200%5C%5C1-N%5E2%20%3D%200%5C%5CN%20%3D%20%5Cpm%201%5C%5CN%20%5Cneq%20-1%5C%5CN%20%3D%201)
Again differentiation Y(N), with respect to N, we get,
![\displaystyle\frac{d^2(Y(N))}{dN^2} = \displaystyle\frac{2kN(N^2-3)}{(1+N^2)^3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%5E2%28Y%28N%29%29%7D%7BdN%5E2%7D%20%3D%20%5Cdisplaystyle%5Cfrac%7B2kN%28N%5E2-3%29%7D%7B%281%2BN%5E2%29%5E3%7D)
At N = 1
Thus, by double derivative test, the maximum value of Y(N) occurs at N = 1.
Thus, largest yield of crop is given by:
Y(1) =
![Y(N) = \displaystyle\frac{k(1)}{1+(1)^2} = \frac{k}{2}](https://tex.z-dn.net/?f=Y%28N%29%20%3D%20%5Cdisplaystyle%5Cfrac%7Bk%281%29%7D%7B1%2B%281%29%5E2%7D%20%3D%20%5Cfrac%7Bk%7D%7B2%7D)