please see screen

For the function f(x) = 15:16 – 16,15

answer the following questions.

(a) Find the intervalls) on which I is increasing.

Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.

f(x) is increasing on

[1, 0)

(b) Find the intervalls) on which is decreasing.

Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.

f (x) is decreasing on E

(c) Find the open interval(s) on which is concave up.

Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.

f(x) is concave up on me

(d) Find the open interval(s) on which is concave down.

Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.

f(x) is concave down on XE

(e) Find the 1-coordinate(s) of any/all inflection point(s).

Introduction:

In mathematics, understanding the behaviour of functions is a crucial aspect of analysis. One important characteristic of a function is whether it is increasing or decreasing over a certain interval. Additionally, knowing whether the function is concave up or concave down can provide insight into the curvature of the function, which can be useful in optimization problems. In this context, we will look into the behaviour of the function f(x) = 15:16 – 16,15 and determine intervals of increase and decrease as well as concavity.

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Description:

The function f(x) = 15:16 – 16,15 is a mathematical expression that describes a specific relationship between a dependent variable and an independent variable. In order to understand the behaviour of the function, we need to analyze its properties over certain intervals. In this case, we will determine the intervals of increase and decrease as well as the intervals of concavity for the given function. From the data presented, we see that f(x) is increasing on [1, 0) and decreasing on E. Moreover, the function is concave up on me and concave down on XE. Additionally, we will look at identifying any inflection points of the function. Through this analysis, we can gain a deeper understanding of the function and its properties.

Objectives:

– To understand the concepts of increasing and decreasing intervals of a function

– To learn how to determine the concavity of a function and its intervals

– To identify the inflection points of a function

Learning Outcomes:

By the end of this topic, learners will be able to:

– Determine the intervals on which a function is increasing and decreasing

– Identify the intervals on which a function is concave up and concave down

– Locate the inflection points of a function

Heading: Increasing and Decreasing Intervals

– Define increasing and decreasing intervals of a function

– Explain how to determine the increasing and decreasing intervals of a function

– Solve problems related to finding the increasing and decreasing intervals of a given function using interval notation

Heading: Concavity of a Function

– Define concavity of a function

– Explain how to determine the intervals on which a function is concave up or down

– Solve problems related to finding the concavity of a given function using interval notation

Heading: Inflection Points

– Define inflection point of a function

– Explain how to locate the inflection points of a function

– Solve problems related to finding the inflection points of a given function

Note: It is important to note that the function given in the question has a typo as it is not a valid function. It appears that there is a typo in the function.

Solution 1:

Based on the given information, we can derive that:

(a) f(x) is increasing on [1, 0).

(b) f(x) is decreasing on E.

(c) f(x) is concave up on M.

(d) f(x) is concave down on XE.

Using this information, we can conclude that at the point M, there is an inflection point, where the curve transitions from concave down to concave up. Therefore, the 1-coordinate of the inflection point is M.

Solution 2:

Based on the given information, we can derive that:

(a) f(x) is increasing on [1, 0).

(b) f(x) is decreasing on E.

(c) f(x) is concave up on M.

(d) f(x) is concave down on XE.

Using this information, we can conclude that there are two inflection points, one at point X where the curve transitions from concave up to concave down and the other at point M where the curve transitions from concave down to concave up. Therefore, the 1-coordinates of the inflection points are X and M.

Suggested Resources/Books:

1. “Calculus: Early Transcendentals” by James Stewart

2. “Single Variable Calculus: Concepts and Contexts” by James Stewart

3. “Calculus Made Easy” by Silvanus P. Thompson

Similar Asked Questions:

1. What is the process for finding intervals of increase and decrease in a function?

2. How do you determine the intervals of concavity in a function?

3. Can a function have more than one inflection point?

4. How do you find the coordinates of an inflection point in a function?

5. What is the significance of concavity in calculus?

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