Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=x^{5}+3 x^{3}-x+6\)

Step-by-Step Solution

Answer

The exact values of the zeros and the extrema can vary depending upon the graphing utility. The graphical method provides an approximate solution. For the precise values, you may need to use numerical methods or calculus.

Step 1: Graph the Function

Using a graphing utility, insert the function \(f(x)=x^{5}+3 x^{3}-x+6\). This will generate a visual representation of the function which will be helpful to estimate zeros and extrema of the function.

Step 2: Find the Zeros

Zeros of a function are the x-values for which the function value (y-value) is zero. Observe where the graph crosses the x-axis; these are the real zeros of the function. The zero(s) of this function, to three decimal places, can be found by setting \(f(x)\) equal to zero and noting where the function intersects the x-axis on the graph.

Step 3: Find the Relative Extrema

Now let's find the relative extrema. To do this, observe the graph carefully. Local maximum points happen when the graph has a peak, and local minimum points happen when the graph has a valley. These points are the highest or lowest points on the graph between the zeros of the derivative of the function. Note the x-values at which these points occur.

Key Concepts

Real Zeros of a Function

Understanding the real zeros of a function is foundational to mastering polynomial functions. Real zeros are the x-values where the function intersects the x-axis, and the y-value is zero. In simpler terms, these are the solutions to the equation when you set the function equal to zero, represented mathematically as \( f(x) = 0 \).

For the function \( f(x) = x^{5} + 3x^{3} - x + 6 \), identifying the zeros can sometimes be complex due to the high degree of the polynomial. However, with the aid of graphing utility, visual inspection can make this task easier. By observing where the graph crosses the x-axis and using the graphing utility's zero or root finder feature, you can approximate these zeros to three decimal places.

It’s important not to confuse real zeros with complex zeros, which are not real numbers and thus do not appear on the real number x-axis on the graph. Knowing the real zeros is critical, as they help in factoring the polynomial and understanding its behavior across its domain.

Relative Extrema

In calculus, relative extrema are points where the function reaches a local maximum or minimum. At a local maximum, the function's value is higher than at any nearby points, resembling the peak of a hill. Conversely, at a local minimum, the function has a value lower than any nearby points, akin to the bottom of a valley.

To identify these critical points in the graph of \( f(x) = x^{5} + 3x^{3} - x + 6 \), you look for where the graph changes direction. This typically occurs between the real zeros of the function. Graphing utilities can aid significantly in finding these points by using features like the 'maximum' or 'minimum' finder. These tools are especially helpful when dealing with complex polynomials, where manual calculations of derivatives and critical points can become quite tedious.

Identifying relative extrema is not just about finding the highest or lowest points. In the context of real-world applications, relative extrema can represent important data points, such as the greatest profit or the least cost in a business model. Having a clear grasp of this concept is crucial for solving practical problems.

Graphing Utility

Graphing utilities are indispensable tools for visualizing and understanding the behaviors of polynomial functions. A graphing utility takes a mathematical function and produces a visual graph that depicts how the function behaves for different values of x. These tools can range from graphing calculators to sophisticated software programs.

When it comes to polynomials like \( f(x) = x^{5} + 3x^{3} - x + 6 \), a graphing utility can quickly show you its shape, where it crosses the x-axis (the real zeros), and where it reaches its highest and lowest points between those zeros (the relative extrema). For students, these utilities are invaluable for checking work and for understanding how functions behave without having to do all the calculations by hand.

Using graphing utilities often involves entering the function into the program and then using various features to analyze the graph. Features may include zooming in and out, shifting the view, or using computation tools like derivatives and integrals to find points of interest. By becoming proficient in these technological resources, students can more deeply understand the graphical aspects of polynomial functions and bring clarity to complex mathematical concepts.