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 Numbers of a Function
Given the function \( f(x) = 2x^3 - 36x^2 + 210x - 4 \), it is known that this function has two critical numbers.
To find these critical numbers, you need to determine the values of \( x \) at which the first derivative of the function equals zero, \( f'(x) = 0 \).
**Task:**
- Find the smaller critical number (denote it as \( x \)).
- Find the larger critical number (denote it as \( x \)).
**Answer Boxes:**
- The smaller one is \( x = \) [ ]
- and the larger one is \( x = \) [ ]
### Instructions:
1. **Find the First Derivative:**
Calculate the first derivative of \( f(x) \).
2. **Solve for Critical Points:**
Set the first derivative equal to zero and solve for \( x \).
3. **Identify the Critical Numbers:**
Determine which of these solutions are the smaller and larger critical numbers.
These concepts are fundamental in calculus and are useful in understanding the behavior of functions, such as finding local maxima, minima, and points of inflection.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c845cf8-3d77-4ddd-9d5a-2a560adfc5de%2F0b2b8f84-1452-470f-8dab-2fa7710a76ff%2F0ju3hwg_processed.jpeg&w=3840&q=75)
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