A model for the US average price of a pound of white sugar...

A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function $$\begin{aligned} \mathrm{S}(\mathrm{t})=&-0.00003337 \mathrm{t}^{5}+0.0009037 \mathrm{t}^{4}-0.008956 \mathrm{t}^{3} \\ &+0.0362 \mathrm{qt}^{2}-0.04458 \mathrm{t}+0.4074 \end{aligned}$$ where t is measured in years since August of $1993 .$ Estimate the times when sugar was cheapest and most expensive during the period $1993-2003 .$

A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function
$$\begin{aligned} \mathrm{S}(\mathrm{t})=&-0.00003337 \mathrm{t}^{5}+0.0009037 \mathrm{t}^{4}-0.008956 \mathrm{t}^{3} \\ &+0.0362 \mathrm{qt}^{2}-0.04458 \mathrm{t}+0.4074 \end{aligned}$$
where t is measured in years since August of $1993 .$ Estimate the times when sugar was cheapest and most expensive during the period $1993-2003 .$

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Key Concepts

Calculus - Derivatives

The derivative of a function represents its rate of change with respect to a variable. By taking derivatives, one can identify points where the function’s slope is zero—indicating potential local maximums or minimums—thereby aiding in understanding the behavior of the function over time.

Optimization on Closed Intervals

Optimization on closed intervals involves evaluating a function at its critical points and at the boundaries of the interval. This method ensures that the absolute maximum or minimum values of the function within a specific domain are correctly identified.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential in determining where a function may have local extrema, which is crucial when looking for the highest or lowest values in a given interval.

Polynomial Functions

Polynomial functions are mathematical expressions involving terms in the form of powers of a variable. They are used to model relationships between quantities, allowing for analysis of trends and patterns over a range, such as time. These functions can be fitted to data, and their behavior, especially over a specified interval, can be studied using calculus.

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