Find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. f(x) = In(9 – In x) Critical numbers: (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) X = Increasing: (Select all that apply.) O (-∞, 0) O (e°, ∞) | (0, eº) O (-∞, ∞) O none of these Decreasing: (Select all that apply.) O (-∞, 0) O (e°, ∞) O (0, eº) (-∞, ∞) O none of these Relative extrema: (If an answer does not exist, enter DNE.) relative maximum (х, у) %3 (x, V) = (| relative minimum
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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**Finding Critical Numbers and Analyzing the Function**
**Objective:**
Find the critical numbers of \( f \) (if any). Identify the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results.
\[ f(x) = \ln(9 - \ln x) \]
**Critical Numbers:**
(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
\[ x = \_\_\_\_\_\_\_\_\_ \]
**Increasing:**
(Select all that apply.)
[ ] \((- \infty, 0)\)
[ ] \((e^9, \infty)\)
[ ] \((0, e^9)\)
[ ] \((- \infty, \infty)\)
[ ] None of these
**Decreasing:**
(Select all that apply.)
[ ] \((- \infty, 0)\)
[ ] \((e^9, \infty)\)
[ ] \((0, e^9)\)
[ ] \((- \infty, \infty)\)
[ ] None of these
**Relative Extrema:**
(If an answer does not exist, enter DNE.)
Relative maximum \((x, y)\) = \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
Relative minimum \((x, y)\) = \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
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**Explanation:**
The aim of this exercise is to determine the behavior of the logarithmic function \(f(x) = \ln(9 - \ln x)\). By finding the critical points, we can establish where the function is increasing or decreasing. The critical points occur where the derivative of the function is zero or undefined.
Once the critical points are determined, the function's behavior can be analyzed over different intervals. Depending on the signs of the derivative in those intervals, the function will be identified either as increasing or decreasing.
This information will help to pinpoint the relative maxima and minima of the function, which are essential in understanding the overall shape and behavior of the curve represented by \(f(x)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7485da11-8b69-4f35-93f2-82155680da51%2F3fac88dc-223f-43cd-8b43-4bfbec9e8273%2Fs7ctcmc_processed.png&w=3840&q=75)
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