Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = e³x + e-x (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) (0,00) Find the interval on which fis decreasing. (Enter your answer using interval notation.) (-∞0,00) (b) Find the local minimum and maximum values of f. local minimum value local maximum value (c) Find the inflection point. (x, y) =([ Find the interval on which fis concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.)
Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = e³x + e-x (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) (0,00) Find the interval on which fis decreasing. (Enter your answer using interval notation.) (-∞0,00) (b) Find the local minimum and maximum values of f. local minimum value local maximum value (c) Find the inflection point. (x, y) =([ Find the interval on which fis concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.)
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.5: Graphical Differentiation
Problem 2E
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![### Calculus Problem: Analyzing the Function \( f(x) = e^{3x} + e^{-x} \)
Consider the equation below. (If an answer does not exist, enter DNE.)
\[ f(x) = e^{3x} + e^{-x} \]
#### (a) Increasing and Decreasing Intervals
1. **Find the interval on which \( f \) is increasing.** (Enter your answer using interval notation.)
\[
(0, \infty)
\]
2. **Find the interval on which \( f \) is decreasing.** (Enter your answer using interval notation.)
\[
(-\infty, 0)
\]
#### (b) Local Minimum and Maximum Values
1. **Find the local minimum value of \( f \).**
\[
\_\_\_\_\_\_\_\_\_
\]
2. **Find the local maximum value of \( f \).**
\[
\_\_\_\_\_\_\_\_\_
\]
#### (c) Inflection Point
1. **Find the inflection point.**
\[
(x, y) = (\_\_\_\_\_\_\_\_\_, \_\_\_\_\_\_\_\_\_)
\]
2. **Find the interval on which \( f \) is concave up.** (Enter your answer using interval notation.)
\[
\_\_\_\_\_\_\_\_\_
\]
3. **Find the interval on which \( f \) is concave down.** (Enter your answer using interval notation.)
\[
\_\_\_\_\_\_\_\_\_
\]
### Graphs and Diagrams
There are no graphs or diagrams provided in the original image. The problem strictly consists of mathematical queries about the properties and characteristics of the given function \( f(x) \).
For future lessons or assignments, consider plotting the function to visually demonstrate the intervals of increase/decrease and concavity, as well as to identify critical points like local minima, maxima, and inflection points. This approach enhances understanding through visual aids alongside the mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80fd0a7-9ae8-4bc5-b926-d69bfc10c0d7%2F6c646d18-bd4a-44e6-9774-8eefd83fd395%2Fi0jx6k9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Analyzing the Function \( f(x) = e^{3x} + e^{-x} \)
Consider the equation below. (If an answer does not exist, enter DNE.)
\[ f(x) = e^{3x} + e^{-x} \]
#### (a) Increasing and Decreasing Intervals
1. **Find the interval on which \( f \) is increasing.** (Enter your answer using interval notation.)
\[
(0, \infty)
\]
2. **Find the interval on which \( f \) is decreasing.** (Enter your answer using interval notation.)
\[
(-\infty, 0)
\]
#### (b) Local Minimum and Maximum Values
1. **Find the local minimum value of \( f \).**
\[
\_\_\_\_\_\_\_\_\_
\]
2. **Find the local maximum value of \( f \).**
\[
\_\_\_\_\_\_\_\_\_
\]
#### (c) Inflection Point
1. **Find the inflection point.**
\[
(x, y) = (\_\_\_\_\_\_\_\_\_, \_\_\_\_\_\_\_\_\_)
\]
2. **Find the interval on which \( f \) is concave up.** (Enter your answer using interval notation.)
\[
\_\_\_\_\_\_\_\_\_
\]
3. **Find the interval on which \( f \) is concave down.** (Enter your answer using interval notation.)
\[
\_\_\_\_\_\_\_\_\_
\]
### Graphs and Diagrams
There are no graphs or diagrams provided in the original image. The problem strictly consists of mathematical queries about the properties and characteristics of the given function \( f(x) \).
For future lessons or assignments, consider plotting the function to visually demonstrate the intervals of increase/decrease and concavity, as well as to identify critical points like local minima, maxima, and inflection points. This approach enhances understanding through visual aids alongside the mathematical analysis.
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