Consider the function f(x) = x> (x- 10). This function has two critical numbers A < B Then A = and B For each of the following intervals, tell whether f(x) is increasing or decreasing. (-0, A]: ? [A, B]: ? [B, c0) ? The critical number A is ? + and the critical number B is ? There are two numbers C < D where either f"(x) = 0 or f"(x) is undefined. Then C = and D = Finally for each of the following intervals, tell whether f(x) is concave up or concave down. (-00, C): ? (C, D) ? (D, 00) ?
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.
![### Analyzing the Function \( f(x) = x^{4/5} (x - 10) \)
This function has two critical numbers \( A < B \).
#### Steps to Analyze the Function:
1. **Determine Critical Numbers:**
- Then \( A = \_\_\_\_ \) and \( B = \_\_\_\_ \).
2. **Intervals for Increasing/Decreasing Behavior:**
- For each of the following intervals, indicate whether \( f(x) \) is increasing or decreasing.
- \( (-\infty, A): \) [Select Increasing/Decreasing]
- \( [A, B]: \) [Select Increasing/Decreasing]
- \( (B, \infty): \) [Select Increasing/Decreasing]
3. **Identify Critical Numbers:**
- The critical number \( A \) is [Select] and the critical number \( B \) is [Select].
4. **Inflection Points or Undefined Second Derivative:**
- There are two numbers \( C < D \) where either \( f''(x) = 0 \) or \( f''(x) \) is undefined.
- Then \( C = \_\_\_\_ \) and \( D = \_\_\_\_ \).
5. **Intervals for Concavity:**
- Finally, for each of the following intervals, indicate whether \( f(x) \) is concave up or concave down.
- \( (-\infty, C): \) [Select Concave Up/Down]
- \( (C, D): \) [Select Concave Up/Down]
- \( (D, \infty): \) [Select Concave Up/Down]
Note that each blank and drop-down must be filled in according to the calculations and analysis of the function's derivatives to determine critical points and concavity behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61dec6a4-dd58-413c-ac38-185006e7681f%2Fc8e818e6-3624-449a-b6cf-e3bc48df3a7a%2Fjwssms6_processed.jpeg&w=3840&q=75)
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