Problem 107 - College Algebra Essentials

Question

use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$ f(x)=x^{3}-6 x^{2}+9 x+1 $$

Step-by-Step Solution

Answer

Without specific values, the exact intervals cannot be provided. However, the solution involved graphing the function, finding its derivative, setting the derivative equal to zero to find critical points, and analyzing the derivative's sign on the intervals determined by the critical points.

Step 1: Graph the Function

Using a graphing utility, plot the function \( f(x) = x^{3}-6 x^{2}+9 x+1 \) over the interval \(x \in [-5,5]\). The function should appear as a curved line, crossing the x-axis at various points.

Step 2: Determine the derivative

Find the derivative of the function, \( f'(x) \). This will be used to find the intervals of increase, decrease, and constants. The derivative is \( f'(x) = 3x^{2} -12x + 9 \).

Step 3: Find the critical points

Set the derivative equal to zero and solve for x to find the critical points: \(0 = 3x^{2} -12x + 9\). The solutions to this equation are the x-values where the graph could potentially change direction (from increasing to decreasing or vice versa).

Step 4: Determine the intervals of increase and decrease

Check the sign of the derivative on the intervals determined by the critical points of the function. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.

Step 5: Analyze the graph

Examine the graph throughout its domain. Clearly read off the intervals of increase, decrease, and constants from the graph. The graph of the function, along with the analysis of derivatives, will guide here.

Key Concepts

Graphing Utility

In today's digital age, graphing utilities have become indispensable tools for students and professionals alike in the field of mathematics. They are computer programs or online apps designed to visually represent equations and functions in a graphical format, making it easier to conceptualize complex mathematical ideas.

For instance, to graph the function given in the exercise, \( f(x) = x^{3} - 6x^{2} + 9x + 1 \), a student can input it into a graphing utility specifying the viewing rectangle through the range \(x \in [-5, 5]\) as instructed. These utilities typically provide a Cartesian plane on which the function will appear as a visual curve. This curve illustrates how the value of the function \( f(x) \) changes with respect to \( x \) within the given range.

Graphing utilities like Desmos, GeoGebra, or even the graphing calculator provided with many handheld calculators not only plot the graph but also often offer additional functionalities. These include zooming in and out for a detailed view, identifying key points on the graph such as intercepts and peaks, and even calculating derivatives at any given point of the function. Utilizing these features, students can better understand the behavior of the function across different intervals, which ultimately leads to a deeper comprehension of the mathematical concepts at hand.

Increasing and Decreasing Intervals

Determining where a function is increasing or decreasing plays a crucial role in understanding the overall behavior of the function. An increasing interval of a function is where its output, \( f(x) \) values, are rising as \( x \) increases. Conversely, a function is said to be decreasing on an interval if the output values fall as \( x \) increases.

When confronted with a function like \( f(x) = x^{3} - 6x^{2} + 9x + 1 \), we can analyze its behavior by first graphing the function, as per our graphing utility. With the graph plotted, one needs to look for slopes where the curve rises and falls. These slopes are a quick visual representation of increasing (upward slope) and decreasing (downward slope) behavior.

However, for a more analytical approach, especially when precision is key, we calculate the function's first derivative, \( f'(x) \), which gives us the slope of the tangent line at any point on the graph. By setting this derivative to zero, we can find critical points that may indicate where the function switches from increasing to decreasing or vice versa. By testing values between these critical points, one can confidently determine the exact intervals of increase and decrease. This understanding is invaluable, as these intervals can have important implications in various scenarios such as optimization problems and understanding the rates of change in applied contexts.

First Derivative Test

The First Derivative Test is a powerful tool in calculus used to determine the local maximum and minimum points on the graph of a function. It involves taking the derivative of the function and analyzing the signs of the derivative before and after critical points.

To illustrate with the solution steps outlined, once the derivative \( f'(x) = 3x^{2} - 12x + 9 \) is obtained, we find its roots, which represent our critical points. If the sign of the derivative changes from positive to negative as we pass through a critical point, this point is identified as a local maximum. If the sign changes from negative to positive, we have a local minimum.
If a sign change does not occur, the critical point may be an inflection point instead of a local extremum. In our particular function, analyzing the signs on either side of the critical points helps determine whether the graph is cresting at a hilltop (local maximum) or rising from a valley (local minimum). This test is particularly valuable for it allows us to navigate the graph and make accurate predictions about the function's local behavior without having to extensively analyze the whole function over its domain.