Determine the intervals on which the function is concave up or down and find the points of inflection. f(x) %3D бх3 — 11х? + 7 (Give your answer as a comma-separated list of points in the form (* , *). Express numbers in exact form. Use symbolic notation and fractions where needed.) points of inflection: Determine the interval on which ƒ is concave up. (Give your answer as an interval in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) Determine the interval on which ƒ is concave down. (Give your answer as an interval in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) хе
Determine the intervals on which the function is concave up or down and find the points of inflection. f(x) %3D бх3 — 11х? + 7 (Give your answer as a comma-separated list of points in the form (* , *). Express numbers in exact form. Use symbolic notation and fractions where needed.) points of inflection: Determine the interval on which ƒ is concave up. (Give your answer as an interval in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) Determine the interval on which ƒ is concave down. (Give your answer as an interval in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) хе
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Concept explainers
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.
Question
![### Concavity and Points of Inflection
#### Problem:
Determine the intervals on which the function is concave up or down and find the points of inflection.
#### Function:
\[ f(x) = 6x^3 - 11x^2 + 7 \]
#### Instructions:
- Provide your answer as a comma-separated list of points in the form \((*, *)\).
- Express numbers in exact form. Use symbolic notation and fractions where needed.
##### Points of Inflection:
- Find the points of inflection by identifying points where the concavity changes.
##### Determine the Interval on which \( f \) is Concave Up:
- Provide your answer as an interval in the form \((*, *)\).
- Use the symbol \( \infty \) for infinity, \(\cup\) for combining intervals.
- Use parentheses "\( (, ) \)", "\([, ]\)" depending on whether the interval is open or closed.
- Enter \(\emptyset\) if the interval is empty.
\[ x \in \]
##### Determine the Interval on which \( f \) is Concave Down:
- Provide your answer as an interval in the form \((*, *)\).
- Use the symbol \( \infty \) for infinity, \(\cup\) for combining intervals.
- Use parentheses "\( (, ) \)", "\([, ]\)" depending on whether the interval is open or closed.
- Enter \(\emptyset\) if the interval is empty.
\[ x \in \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd5995a-c4b2-468d-a13c-4876a73e2cf6%2F5a44adf9-2545-4ebd-903b-592022423442%2Ffan75x_processed.png&w=3840&q=75)
Transcribed Image Text:### Concavity and Points of Inflection
#### Problem:
Determine the intervals on which the function is concave up or down and find the points of inflection.
#### Function:
\[ f(x) = 6x^3 - 11x^2 + 7 \]
#### Instructions:
- Provide your answer as a comma-separated list of points in the form \((*, *)\).
- Express numbers in exact form. Use symbolic notation and fractions where needed.
##### Points of Inflection:
- Find the points of inflection by identifying points where the concavity changes.
##### Determine the Interval on which \( f \) is Concave Up:
- Provide your answer as an interval in the form \((*, *)\).
- Use the symbol \( \infty \) for infinity, \(\cup\) for combining intervals.
- Use parentheses "\( (, ) \)", "\([, ]\)" depending on whether the interval is open or closed.
- Enter \(\emptyset\) if the interval is empty.
\[ x \in \]
##### Determine the Interval on which \( f \) is Concave Down:
- Provide your answer as an interval in the form \((*, *)\).
- Use the symbol \( \infty \) for infinity, \(\cup\) for combining intervals.
- Use parentheses "\( (, ) \)", "\([, ]\)" depending on whether the interval is open or closed.
- Enter \(\emptyset\) if the interval is empty.
\[ x \in \]
![### Calculating Limits Using L'Hôpital's Rule
#### Problem Statement
Consider the limit:
\[
\lim_{x \to 0} \left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right],
\]
which leads to the indeterminate form \(1^\infty\).
#### Methodology
1. **Substitution and Transformation:**
- Set \(y = (\cos(x) + 6x)^{1/(7 \sin(x))}\) and use L'Hôpital’s Rule to find:
\[
\lim_{x \to 0} \ln(y) = \lim_{x \to 0} \ln\left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right].
\]
2. **Finding the Limit of the Logarithmic Form:**
- Calculate the limit:
\[
\lim_{x \to 0} \ln\left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right] = \boxed{\phantom{0}}
\]
3. **Using the Result:**
- Use the above result to find the limit:
\[
\lim_{x \to 0} y = \lim_{x \to 0} \left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right].
\]
4. **Final Calculation:**
- Calculate the final limit:
\[
\lim_{x \to 0} \left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right] = \boxed{\phantom{0}}
\]
#### Explanation
The problem involves finding the limit of an expression that at first glance appears indeterminate. By transforming the expression using logarithms, we apply L'Hôpital's Rule to resolve the indeterminacy and compute the limit.
Note: \(\boxed{\phantom{0}}\) signifies fields where further detailed computation is required to obtain specific numerical results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd5995a-c4b2-468d-a13c-4876a73e2cf6%2F5a44adf9-2545-4ebd-903b-592022423442%2Fqlyp5d_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Limits Using L'Hôpital's Rule
#### Problem Statement
Consider the limit:
\[
\lim_{x \to 0} \left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right],
\]
which leads to the indeterminate form \(1^\infty\).
#### Methodology
1. **Substitution and Transformation:**
- Set \(y = (\cos(x) + 6x)^{1/(7 \sin(x))}\) and use L'Hôpital’s Rule to find:
\[
\lim_{x \to 0} \ln(y) = \lim_{x \to 0} \ln\left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right].
\]
2. **Finding the Limit of the Logarithmic Form:**
- Calculate the limit:
\[
\lim_{x \to 0} \ln\left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right] = \boxed{\phantom{0}}
\]
3. **Using the Result:**
- Use the above result to find the limit:
\[
\lim_{x \to 0} y = \lim_{x \to 0} \left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right].
\]
4. **Final Calculation:**
- Calculate the final limit:
\[
\lim_{x \to 0} \left[(\cos(x) + 6x)^{1/(7 \sin(x))}\right] = \boxed{\phantom{0}}
\]
#### Explanation
The problem involves finding the limit of an expression that at first glance appears indeterminate. By transforming the expression using logarithms, we apply L'Hôpital's Rule to resolve the indeterminacy and compute the limit.
Note: \(\boxed{\phantom{0}}\) signifies fields where further detailed computation is required to obtain specific numerical results.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning