Can a Polynomial Function Have Both an Absolute Maximum and an Absolute Minimum?

Polynomial functions, being continuous and differentiable, often exhibit behaviors like attaining both an absolute maximum and an absolute minimum within certain intervals. This article delves into the mathematical concepts and applications behind this phenomenon, explaining how such extrema can be located and evaluated.

Properties of Polynomial Functions

Polynomial functions are continuous and differentiable everywhere on the real number line. This characteristic is crucial for the existence of maximum and minimum values. By leveraging the properties of continuity and differentiability, we can better understand and locate these extrema in polynomial functions.

The Role of Closed Intervals

To find the absolute extrema (both the absolute maximum and minimum) within a function, we typically consider the function on a closed interval [a, b]. The Extreme Value Theorem guarantees that a continuous function on a closed interval will attain both its maximum and minimum values at least once within that interval. This theorem is the foundational principle for our analysis.

Locating Critical Points

Identifying the critical points of a polynomial function is a crucial step in finding its maximum and minimum values. A critical point is where the function’s derivative is zero or undefined. Once identified, these points can be used to evaluate the function and determine the extrema. Here are the steps involved:

tTake the derivative of the polynomial function. tSet the derivative equal to zero to find the critical points. tEvaluate the function at these critical points to obtain potential maximum and minimum values. tDo not forget to evaluate the function at the endpoints of the interval, as the maximum and minimum values could also occur at these points.

By comparing the values obtained from the critical points and the endpoints, we can identify the absolute maximum and minimum.

An Example Demonstration

Consider the polynomial function f(x) -x^2 4 over the closed interval [-3, 3]. Let’s walk through the steps to find the absolute maximum and minimum values.

tEvaluate the function at the endpoints: t ttf(-3) -(-3)^2 4 -9 4 -5 ttf(3) -(3)^2 4 -9 4 -5 t tFind the critical points by taking the derivative and setting it to zero: t ttf'(x) -2x tt-2x 0 implies x 0 t tEvaluate the function at the critical point: t ttf(0) -(0)^2 4 4 t tCompare the values at the critical points and endpoints: t ttf(-3) -5 ttf(0) 4 ttf(3) -5 t tFrom these values, we can see that: t ttThe absolute maximum value is 4, occurring at x 0. ttThe absolute minimum value is -5, occurring at both x -3 and x 3. t

Conclusion

In summary, a polynomial function can indeed have both an absolute maximum and an absolute minimum when evaluated over a closed interval. This outcome is a direct consequence of the function's continuity and the application of the Extreme Value Theorem. By understanding and applying these principles, one can effectively locate and evaluate these critical points, ensuring a comprehensive analysis of polynomial functions.

References:

MathIsFun - Polynomials

Khan Academy - Extreme Values Review