A variation of the von Liebig model states that the yield $f(x)$ of a plant, measured in bushels, responds to the amount $x$ of potassium in a fertilizer according to the following square root model: $$f(x) =-0.057-0.417x+0.852\sqrt{x}$$

For what amounts of potassium will the yield increase? For what amounts of potassium will the yield decrease?

Solution The yield is increasing when $f^\prime (x)>0.$ $$f^\prime (x) =-0.417+\dfrac{0.426}{\sqrt{x}}=\dfrac{-0.417\sqrt{x}+0.426}{\sqrt{x}}$$

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Now, $f^\prime (x) >0$ when $$\begin{array}{l} - 0.417\sqrt x + 0.426 > 0 \\ \,\,\,0.417\sqrt x - 0.426 < 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt x < 1.022 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt x < 1.044 \\ \end{array}$$

The crop yield is increasing when the amount of potassium in the fertilizer is less than $1.044$ and is decreasing when the amount of potassium in the fertilizer is greater than $1.044$.

*Source: Quirino Paris. (1992). The von Liebig Hypothesis. American Journal of Agricultural Economics, 74(4), 1019–1028.