Biology Ring and Benedict ^30 created a mathematical model...

Biology Ring and Benedict $^{30}$ created a mathematical model of the yield response of cotton to injury by the bollworm. Normalized yield $y$ (yield with injury divided by yield without injury) was shown to be approximated by $\ln y=0.1494 \ln x-5.5931 x+7.12 x^{2},$ where $x$ is the number of injured reproductive organs (flower buds plus capsules) per $100 .$ Graph on your grapher, and determine the approximate value of $x$ for which $y$ attains a maximum on the interval [0,0.05] . Confirm, using calculus.

Biology Ring and Benedict $^{30}$ created a mathematical model of the yield response of cotton to injury by the bollworm. Normalized yield $y$ (yield with injury divided by yield without injury) was shown to be approximated by $\ln y=0.1494 \ln x-5.5931 x+7.12 x^{2},$ where $x$ is the
number of injured reproductive organs (flower buds plus capsules) per $100 .$ Graph on your grapher, and determine the approximate value of $x$ for which $y$ attains a maximum on the interval [0,0.05] . Confirm, using calculus.

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Key Concepts

Graphical Analysis

Graphical analysis involves plotting the function over a given interval to visually inspect the behavior of the function. By graphing, one can identify approximate locations where the function attains a maximum or minimum, which serves as a preliminary check before more rigorous analytic methods are applied.

Calculus-Based Optimization

Calculus-based optimization uses derivatives to determine the critical points of a function. By differentiating the function with respect to the variable, setting the derivative equal to zero, and solving for the variable, one can find potential points where the function may have a maximum or minimum. This method provides an exact algebraic confirmation of the results suggested by graphical analysis.

Monotonic Transformations

A monotonic transformation is a function that preserves the order of the values it is applied to. In this context, taking the natural logarithm of the yield means that maximizing the logarithm of the yield is equivalent to maximizing the yield itself. This property simplifies the optimization process without altering the location of the maximum point.

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