2 Find the critical points of the function f(x) = x³ + x² + 40x + 2. Use the First Derivative Test to determine whether the critical point is a local minimum or local maximum (or neither). (Use symbolic notation and fractions where needed. Give your answers in the form of comma separated lists. Enter DNE if there are no critical points.) f has local a minimum at f has a local maximum at Find the intervals on which the given function is increasing or decreasing. (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parentheses "(",")", "[", or "]" depending on whether the interval is open or closed.) the function is increasing on the function is decreasing on
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.
![### Critical Points and Intervals Analysis of the Function
#### Problem Statement
Consider the function \( f(x) = \frac{1}{3}x^3 + \frac{13}{2}x^2 + 40x + 2 \). Use the First Derivative Test to determine whether the critical point is a local minimum or local maximum (or neither).
#### Instructions
- Use symbolic notation and fractions where needed.
- Give your answers in the form of comma-separated lists.
- Enter DNE if there are no critical points.
### Determine Local Minima and Maxima
- \( f \) has a local minimum at: \_\_\_\_\_\_\_\_\_
- \( f \) has a local maximum at: \_\_\_\_\_\_\_\_\_
### Find Increasing and Decreasing Intervals
Identify the intervals on which the given function is increasing or decreasing.
- Use symbolic notation and fractions where needed.
- Give your answers as intervals in the form \((\ast, \ast)\).
- Use the symbol \(\infty\) for infinity.
- Use \(\cup\) for combining intervals and an appropriate type of parentheses \("("," ")", "[", "\]" depending on whether the interval is open or closed.
- The function is increasing on: \_\_\_\_\_\_\_\_\_
- The function is decreasing on: \_\_\_\_\_\_\_\_\_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95425ac5-3f57-4b1e-926d-91b4503a2e10%2F3fe9ed4f-cac3-4801-ab37-7520fd3f1834%2F89noreb.png&w=3840&q=75)
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