What is the domain? Use a graphing utility to determine the intervals in which the function is increasing and decreasing and approximate any relative maximum or minimun values of the function.
g(x) = 12 Ln x / x
My answer was: Domain = (12,0)
decreasing = (3,-2) and increasing = (-2,3) but these were wrong.
To determine the domain of a function, you need to consider any restrictions on the input values. In the case of the function g(x) = (12 ln x) / x, the only restriction is that x must be greater than 0. This is because the natural logarithm function ln(x) is only defined for positive numbers.
So, the domain of g(x) is x > 0 or (0, +โ).
To determine the intervals where the function g(x) is increasing or decreasing, you can use a graphing utility. One popular graphing utility is Desmos (www.desmos.com).
1. Open Desmos or any other graphing utility.
2. Enter the function g(x) = (12 ln x) / x into the graphing utility.
3. Observe the graph of the function.
In the graph, you will notice that the function g(x) is always positive since both ln x (natural logarithm) and x are positive for x > 0. Additionally, as x approaches 0 from the right (x โ 0+), g(x) approaches positive infinity. This means that the function does not have a relative maximum or minimum value.
Therefore, the correct answers are:
- Domain: x > 0 or (0, +โ).
- The function g(x) is always increasing on the interval (0, +โ).
Hope this helps! Let me know if you have any further questions.