Problem 90 - Calculus of a Single Variable

Question

Creating Polynomial Functions In Exercises \(87-90,\) find a polynomial function $$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0} $$ that has only the specified extrema. (a) Determine the minimum degree of the function and give the criteria you used in determining the degree. (b) Using the fact that the coordinates of the extrema are solution points of the function, and that the \(x\) -coordinates are crition yields the coefficients system of linear equation. (c) Use a graphing utility to solve the system of equations and determine the function. (d) Use a graphing utility to confirm your result graphically. Relative minimum: \((1,2) ;\) Relative maxima: \((-1,4),(3,4)\)

Step-by-Step Solution

Answer

The polynomial function is obtained by first determining the minimum degree of the function using the number of extrema. The extrema values are then used to form a system of equations which can be solved using a graphing utility to obtain the coefficients of the polynomial function. Lastly, a graphical confirmation is done to verify the derived polynomial function.

Step 1: Determine the Degree

The extrema given consists of 1 minimum and 2 maxima which make up total of 3 extrema. A function can have exactly \(n - 1\) extrema for a polynomial of degree \(n\). Therefore, the minimum degree of the polynomial is 4.

Step 2: System of Linear Equations

With the degree 4, the general form of the function is \(f(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}\). The function, at the extremes is equal to their respective \(y\) values. Thus this yields the following system of equations based on extrema: \[\begin{align*}a_{4}(1)^4 + a_{3}(1)^3 + a_{2}(1)^2 + a_{1}(1) + a_{0} &= 2 \a_{4}(-1)^4 + a_{3}(-1)^3 + a_{2}(-1)^2 + a_{1}(-1) + a_{0} &= 4 \a_{4}(3)^4 + a_{3}(3)^3 + a_{2}(3)^2 + a_{1}(3) + a_{0} &= 4\end{align*}\]

Step 3: Solving the System Using a Graphical Utility

We need to solve our system of equations using a graphing calculator or an online graphing tool. Solving the system will give us the coefficients \(a_4, a_3, a_2, a_1, a_0\). These coefficients complete the polynomial function.

Step 4: Graph Confirmation

With the polynomial function derived from the coefficients, we can confirm the result using a graphing utility tool. The function should show the provided extrema.

Key Concepts

Minimizing Polynomial Degree

Determining the minimum degree of a polynomial is crucial when we want a function with specific characteristics, such as given extrema. In essence, for a polynomial function to display a certain number of extrema — those points where the function takes a local maximum or minimum value — the degree of the function should be as minimal as possible yet sufficient to accommodate those extrema.

Consider a polynomial function with one or more local maxima and minima. The relationship between the degree of the function, denoted by 'n', and the number of extrema is expressed by the formula: the maximum number of extrema is one less than the degree of the function, or in other words, a polynomial function of degree 'n' can have up to 'n-1' extrema. Therefore, if we have three extrema, the polynomial must be at least of degree four to exist.

When minimizing the polynomial degree, it is not only about meeting the minimum requirements but also ensuring that we're not introducing unnecessary complexity into the function. A higher degree polynomial means more turning points and perhaps additional unintended extrema. Thus, choosing the minimum degree necessary according to the number of extrema helps to create a simpler, more manageable polynomial function.

Solving System of Linear Equations

In the context of finding the coefficients of a polynomial function given its extrema, we transform the problem into solving a system of linear equations. Here's the logic: at each extremum, the value of the polynomial function must match the given y-coordinate of the extremum. With the polynomial of a certain degree, say four, we can write the function as
\(f(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0\).

Each extremum (x,y) will then provide us with an equation by substituting 'x' and 'y' into the polynomial expression. Collectively, the extrema yield a set of equations where the unknowns are the coefficients of the polynomial. This set is a system of linear equations because each equation is linear with respect to the coefficients despite the function itself being polynomial with respect to 'x'.

These systems are often solved using methods such as substitution, elimination, or matrix techniques like Gaussian elimination or utilizing the inverse matrix. Solving this system gives us the exact values of the coefficients, which define the polynomial uniquely.

Graphing Utility Solutions

Graphing utilities, both in handheld calculators and online software, are indispensable tools for visualizing and solving problems in mathematics, particularly when dealing with polynomial functions. For our specific problem, after establishing a system of linear equations that dictates the coefficients of the polynomial, we may turn to a graphing utility to find these coefficients.

Modern graphing utilities can solve systems of equations either through algebraic manipulations or numerical methods. They allow for visual confirmation of the results by plotting the polynomial function and checking against the given extrema. Moreover, graphical solutions provide a means to comprehend the behavior of the function better and ensure that no additional extrema are present beyond those required.

Graphing utilities can also be programmed to numerically calculate the roots or optima, acting as a solver that complements the analytical methods learned in class. This makes them a reliable resource for checking one's work and gaining a graphical perspective of algebraic problems.

Polynomial Function Coefficients

Polynomial function coefficients are the numbers (\(a_n, a_{n-1}, ... , a_0\)) that multiply the variable raised to various powers in a polynomial expression. Each coefficient impacts the shape and position of the function's graph. Understanding how coefficients affect a polynomial is essential to constructing a function to meet specific criteria, such as prescribed extrema.

For instance, the leading coefficient (associated with the highest power of 'x') affects the width of the graph and determines whether the function's ends rise or fall. The constant term (\(a_0\)) vertically shifts the entire graph. Intermediate coefficients can influence the tilt, curvature, and the location of turning points of the graph. In our exercise, the coefficients are calculated to ensure that the polynomial curve passes through the specific points where the extrema are located.

A change in any coefficient can result in a significant change in the polynomial's graph, which might introduce more extrema or disrupt the existing ones. Therefore, when we find the coefficients through solving a system of linear equations given certain conditions, we're shaping the polynomial function to exhibit the required behavior at specific points.