Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$g(x)=6 x \ln x$$

Step-by-Step Solution

Answer

The domain of the function is \(x > 0\). The function is decreasing on the interval \(0 < x < 1/e\) and increasing on the interval \(x > 1/e\). The function has a relative minimum value at \(x = 1/e\) and the corresponding \(g(x)\) value is \(-6/e \approx -2.202\), The function has no relative maximum.

Step 1: Graph the Function

First, you need to input the function \(g(x)=6x\ln(x)\) into a graphing utility. The graph will give a visual idea of how the function behaves for different values of \(x\). It aids in understanding the function.

Step 2: Find the Domain

The domain of a function is the set of all possible input values (in this case, \(x\)) which will make the function 'work', and will output real numbers. In the function \(g(x)=6x\ln(x)\), as the logarithm function is undefined for \(x \leq 0\), the domain of this function is \(x > 0\)

Step 3: Identify Increasing and Decreasing Intervals

From the graph of the function, the derivative \(g'(x)\) can be calculated to find the intervals where the function is increasing or decreasing. The derivative \(g'(x) = 6\ln(x) + 6\). This derivative is greater than 0 for \(x > 1/e\), therefore the function is increasing for \(x > 1/e\). For \(0< x < 1/e\), \(g'(x) < 0\) and the function is decreasing.

Step 4: Approximate Relative Maximum or Minimum Values

By looking at the graph and from Step 3, it's clear that there are no relative maximum values. The relative minimum value appears at \(x = 1/e\), which is where the function changes from decreasing to increasing. To find the precise minimum value, substitute \(x = 1/e\) into the function \(g(x)\) which gives \(g(1/e) = -6/e \approx -2.202\).

Key Concepts

Increasing and Decreasing Intervals

When analyzing functions like the logarithmic function in our exercise, understanding the increasing and decreasing intervals provides insight into the function's growth and decay. For the function \(g(x)=6x\ln(x)\), this behavior is determined by its derivative \(g'(x)\).

When the derivative of a function is positive (\(g'(x) > 0\)), the function is increasing. Conversely, when the derivative is negative (\(g'(x) < 0\)), the function is decreasing. In the solved exercise, \(g'(x) = 6\ln(x) + 6\), indicating that \(g(x)\) is increasing for \(x > 1/e\) and decreasing for \(0< x < 1/e\). This simple but powerful analysis helps us visualize the function's slope and predict its behavior over different intervals.

Function Domain

The domain of a function is fundamental to understanding the limits within which the function can operate. In more simple terms, it's the set of all possible input values that will result in real numbers after the function is applied.

For logarithmic functions, such as \(g(x)=6x\ln(x)\), we must be mindful that the natural logarithm (\ln) is not defined for values of \(x\) that are less than or equal to zero. Therefore, the domain for \(g(x)\) is \(x > 0\). When using graphing utilities or analyzing functions, always consider the domain first as it dictates the functional ‘playing field’ and prevents misunderstanding of the function's properties.

Relative Extrema

The peaks and troughs of a function are known as relative extrema; these can be relative maxima (high points) or relative minima (low points). Finding these points helps us understand where the function reaches its local highs or lows.

In the provided solution, the derivative test reveals that there are no relative maxima for the function \(g(x)=6x\ln(x)\), while a relative minimum is spotted at \(x = 1/e\). At this point, the function switches from decreasing to increasing, indicating a trough. The calculated minimum value of \(g(1/e)\) rounds to approximately -2.202, offering not just a visual cue on the graph but also a quantitative measure of the function's behavior.

Graphing Utilities

Graphing utilities are invaluable tools that help us visualize complex functions, such as logarithmic functions. By feeding the function's formula into a graphing utility, we obtain a graph that represents the function’s behavior across different values, making it easier to comprehend visually.

In our problem, using a graphing utility helped confirm the domain and identify increasing as well as decreasing intervals and relative minimum value for \(g(x)=6x\ln(x)\). It’s an effective visual aid for students to cross-verify the analytical work they do and spot details that might be missed through calculation alone. Always take advantage of such technological tools to enhance your understanding of mathematical concepts.