Heating costs In colder climates the cost for natural gas to...

Heating costs In colder climates the cost for natural gas to heat homes can vary from one month to the next. The polynomial function f(x)=-0.1213 x^{4}+3.462 x^{3}-29.22 x^{2}+64.68 x+97.69 models the monthly cost in dollars of heating a typical home. The input $x$ represents the month, where $x=1$ corresponds to January and $x=12$ to December. (Source: Minnegasco.) (a) Where might the absolute extrema occur for $1 \leq x \leq 12 ?$ $\Rightarrow$ (b) Approximate the absolute extrema and interpret the results.

Heating costs In colder climates the cost for natural gas to heat homes can vary from one month to the next. The polynomial function
f(x)=-0.1213 x^{4}+3.462 x^{3}-29.22 x^{2}+64.68 x+97.69
models the monthly cost in dollars of heating a typical home. The input $x$ represents the month, where $x=1$ corresponds to January and $x=12$ to December. (Source: Minnegasco.)
(a) Where might the absolute extrema occur for $1 \leq x \leq 12 ?$
$\Rightarrow$ (b) Approximate the absolute extrema and interpret the results.

Key Concepts

Critical Points

Critical points are points in the domain of a function where the derivative is zero or undefined. These points are potential candidates for local and absolute extrema. By finding and evaluating the function at these critical points, along with the boundary points of the domain, one can determine where the function achieves its highest and lowest values.

Interpretation of Modeling Results

When a function models a real-world scenario, such as heating costs over time, interpreting the absolute extrema provides valuable insights. The absolute maximum might indicate the month of highest cost, while the absolute minimum identifies the month with the lowest cost. This insight is useful for planning and decision-making, as it directly relates mathematical outcomes to practical implications.

Polynomial Function

A polynomial function is an expression that involves only non-negative integer powers of the variable. In modeling contexts, these functions can be used to represent trends and fluctuations in data over an interval. Their smooth and continuous nature makes them particularly useful for approximating real-world phenomena where variables change gradually over time.

Absolute Extrema in a Closed Interval

Absolute extrema refer to the highest or lowest function values on a given closed interval. The Extreme Value Theorem guarantees that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum on that interval. This concept is central when analyzing cost models or real-world scenarios, ensuring that one can identify the overall peaks and troughs.

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