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 $$ \begin{aligned} S(t)=- & 0.00003237 t^5+0.0009037 t^4-0.008956 t^3 \\ & +0.03629 t^2-0.04458 t+0.4074 \end{aligned} $$ 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
$$
\begin{aligned}
S(t)=- & 0.00003237 t^5+0.0009037 t^4-0.008956 t^3 \\
& +0.03629 t^2-0.04458 t+0.4074
\end{aligned}
$$
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.

Key Concepts

Global vs Local Extrema

Local extrema are points where the function attains a minimum or maximum value in a small neighborhood, while global (or absolute) extrema are the highest or lowest values over the entire interval under consideration. Differentiating between the two is important because in restricted domains, the absolute extremes can occur at either critical points or endpoints.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients organized in nonnegative integer powers. Understanding polynomial functions is crucial because they can model a variety of real world phenomena, and their smooth, continuous nature allows for the use of calculus-based methods to analyze behavior such as increases, decreases, and curvature over the domain.

Derivatives and Critical Points

The derivative of a function measures its instantaneous rate of change, and setting the derivative equal to zero helps locate critical points. Critical points occur where the function’s behavior may shift from increasing to decreasing (or vice versa), which is key to identifying possible local minima or maxima. This technique is fundamental in optimization problems across many disciplines.

Optimization on Closed Intervals

When optimizing a function over a closed interval, one must consider both the critical points found within the interval and the values at the endpoints. This is because the global maximum or minimum may occur at the boundary of the interval. Evaluating all these points ensures that the absolute extrema are correctly identified.

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