Problem 84 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show that a polynomial function defined on the interval \((-\infty, \infty)\) cannot have both an absolute maximum and an absolute minimum unless it is a constant function.

Step-by-Step Solution

Answer

A non-constant polynomial function has at least one non-zero term in its first derivative, implying that its graph cannot be parallel to the x-axis indefinitely. Therefore, the function cannot be bounded on the interval \((-\infty, \infty)\), making it impossible to have both an absolute maximum and an absolute minimum. Conversely, a constant function has a horizontal graph and retains its maximum and minimum values over the entire domain, satisfying the condition. Thus, a polynomial function defined on the interval \((-\infty, \infty)\) cannot have both an absolute maximum and an absolute minimum unless it is a constant function.

Step 1: Understanding polynomial functions

Polynomial functions are smooth, continuous functions that have the form: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x^1 + a_0\), where \(n\) is a non-negative integer and \(a_n, a_{n-1}, \dots , a_1, a_0\) are constants.

Step 2: Investigate the derivatives of a polynomial function

Derivatives help us to find extremum points of a function. Let's compute the first and second derivatives to understand where the function can have local maximum and minimum points: First derivative: \(f'(x) = na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + a_1\) Second derivative: \(f''(x) = n(n-1)a_nx^{n-2} + (n-1)(n-2)a_{n-1}x^{n-3} + \cdots + 0\)

Step 3: Analyze the behavior of the polynomial function

A local maximum or minimum occurs when the first derivative equals zero, and the second derivative test helps us verify if it's a maximum or minimum. For a non-constant polynomial function with degree \(n>0\), the first derivative will always have at least one non-zero term, regardless of the value of \(x\). Meaning, the polynomial function will present a non-zero tangent, and its graph will not be parallel to the x-axis indefinitely.

Step 4: Prove that a non-constant polynomial function cannot have both an absolute maximum and an absolute minimum

If a non-constant polynomial function had both an absolute maximum and an absolute minimum, the tangent would have to be horizontal at these two points. However, we have seen in Step 3 that the first derivative of the polynomial function (slope of the tangent line) will never be zero indefinitely. Consequently, the graph of the polynomial function either extends indefinitely up or down as the value of \(x\) approaches \(\infty\) or \(-\infty\). The polynomial function will not be bounded on the interval \((-\infty, \infty)\), and it is impossible to have both an absolute maximum and an absolute minimum for a non-constant function.

Step 5: Understand that a constant function satisfies the condition of having both an absolute maximum and an absolute minimum

A constant function is a polynomial function of degree 0, defined as \(f(x) = a_0\), where \(a_0\) is a constant. Its first derivative is 0, which means the function is horizontal and remains the same over its entire domain. Therefore, the constant function retains its maximum and minimum value over the range \((-\infty, \infty)\). In conclusion, a polynomial function defined on the interval \((-\infty, \infty)\) cannot have both an absolute maximum and an absolute minimum unless it is a constant function.

Key Concepts

Extremum Points in Calculus

Understanding extremum points is crucial in calculus. They represent the peaks (maximums) and troughs (minimums) on a graph of a function. To find these points for a polynomial function, we look for values where the function changes direction. This typically occurs where the derivative—essentially the slope of the tangent line to the graph—equals zero.

However, not all points where the derivative is zero are extremum points. To confirm, we may also check the second derivative, following the second derivative test. If the second derivative is positive at a point where the first derivative is zero, the function has a local minimum there. If it's negative, there is a local maximum. If the second derivative is also zero, the test is inconclusive, and further analysis is required.

A polynomial function of degree one or higher can have multiple extremum points or none, but it cannot have both an absolute maximum and an absolute minimum unless it is constant. This is because non-constant polynomial functions are either always increasing or always decreasing as they approach infinity in either direction, hence no bound can be put on their maxima and minima globally.

Derivative of Polynomial Functions

The derivative of a polynomial function gives us the slope of the tangent line at any point along the function. To differentiate a polynomial function, we use the power rule. For each term, we multiply the coefficient by the power of the variable, and then decrease the power by one.

For example, the derivative of the term would be <2a_2x^1> or simply <2a_2x>. Applying this rule term by term gives us the full derivative of the polynomial function. Remember, the derivative is a function itself—it tells us how the original function is changing at every point, providing critical information about the function's behavior, including where its extremum points might be located. Derivatives also reveal inflection points, where the curve changes from concave to convex or vice versa, which are vital for a deeper understanding of a function's shape.

Behavior of Polynomial Functions

The behavior of polynomial functions is determined by their degree and their leading coefficient. The degree tells us the maximum number of solutions, or x-intercepts, and the maximum number of turns, or changes in direction, that the graph can have.

• If the leading coefficient is positive and the degree is even, the polynomial will rise to positive infinity on both ends of the graph.
• If the leading coefficient is negative and the degree is even, the graph falls to negative infinity on both ends.
• With an odd degree, if the leading coefficient is positive, the polynomial will fall to negative infinity to the left and rise to positive infinity to the right.
• If the leading coefficient is negative and the degree is odd, the graph rises to positive infinity to the left and falls to negative infinity to the right.

Knowing the degree and the leading coefficient also helps us to determine the end behavior of the graph, especially as the value of x becomes very large or very small—towards positive or negative infinity. Critical points, which include extremum points and inflections, help to define the graph's shape between these ends. Explaining these concepts thoroughly enhances the understanding of the behavior of polynomial functions.