Determine the intervals on which the graph of y = f(x) is concave up or concave down, and find the points of inflection. f(x) = (x² – 4) e* Provide intervals in the form (*, *). Use the symbol co for infinity, u for combining intervals, and an appropriate type of parenthesis "(", ")", "[", or "]", depending on whether the interval is open or closed. Enter Ø if the interval is empty. Provide points of inflection as a comma-separated list of (x, y) ordered pairs. If the function does not have any inflection points, enter DNE. Use exact values for all responses. f is concave up when x E f is concave down when x E points of inflection:
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.
![**Determine the Intervals of Concavity and Find Points of Inflection**
Consider the function:
\[ f(x) = (x^2 - 4) e^x \]
**Instructions:**
1. **Intervals of Concavity:**
- Provide intervals in the format \(( *, * )\).
- Use the symbol \(\infty\) for infinity and \(\cup\) for the union of intervals.
- Use the appropriate type of parentheses, \(( )\) or \([ ]\), indicating open or closed intervals, respectively.
- If an interval is empty, enter \(\emptyset\).
2. **Points of Inflection:**
- List points of inflection as a comma-separated list of \((x, y)\) ordered pairs.
- If the function has no inflection points, enter DNE.
Use exact values for all responses.
- **\( f \) is concave up when \( x \in \):** [Input Box]
- **\( f \) is concave down when \( x \in \):** [Input Box]
- **Points of inflection:** [Input Box]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd5995a-c4b2-468d-a13c-4876a73e2cf6%2Ff8d11ab4-ba66-428b-b8f0-77fd5f14aa05%2Fpjkupwc_processed.png&w=3840&q=75)
![**Determine the intervals on which the graph of \( y = f(x) \) is concave up or concave down, and find the x-values at which the points of inflection occur.**
\[ f(x) = x(x - 7\sqrt{x}), \quad x > 0 \]
*(Enter an exact answer. Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list, if necessary. Enter DNE if there are no points of inflection.)*
\[ x = \]
---
*(Use symbolic notation and fractions where needed. Give your answers as intervals in the form \( (\ast, \ast) \). Use the symbol \(\infty\) for infinity, \( U \) for combining intervals, and an appropriate type of parenthesis "(", ")", "[", or "]", depending on whether the interval is open or closed. Enter \(\varnothing\) if the interval is empty.)*
\[ f \text{ is concave up when } x \in \]
\[ f \text{ is concave down when } x \in \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd5995a-c4b2-468d-a13c-4876a73e2cf6%2Ff8d11ab4-ba66-428b-b8f0-77fd5f14aa05%2Fwj8dvzv_processed.png&w=3840&q=75)
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