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?
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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.