Find the local maximum and minimum values of f using both the First and Second Derivative Tests. f(x) = 2 + 9x² – 6x3 local maximum value local minimum value
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.
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![### Locating Local Extremes Using Derivative Tests
#### Objective:
Determine the local maximum and minimum values of the function \( f(x) \) by applying both the First and Second Derivative Tests.
#### Given Function:
\[ f(x) = 2 + 9x^2 - 6x^3 \]
#### Instructions:
1. **First Derivative Test:** Identify critical points by differentiating the function and setting the first derivative equal to zero.
2. **Second Derivative Test:** Determine the concavity at the critical points by evaluating the second derivative of the function.
#### Steps to Follow:
1. **Find the first derivative \( f'(x) \):**
\[ f'(x) = \frac{d}{dx}(2 + 9x^2 - 6x^3) \]
2. **Set \( f'(x) \) to zero and solve for \( x \):**
Determine the critical points where \( f'(x) = 0 \).
3. **Find the second derivative \( f''(x) \):**
\[ f''(x) = \frac{d^2}{dx^2}(2 + 9x^2 - 6x^3) \]
4. **Evaluate \( f''(x) \) at the critical points:**
Use the Second Derivative Test to classify each critical point as a local maximum or minimum.
##### Graphs and Diagrams:
- **Local Maximum Value**: Use the results from the First and Second Derivative Tests to input the local maximum value in the provided box.
- **Local Minimum Value**: Similarly, input the local minimum value in the accompanying box.
#### Interaction Fields:
- **local maximum value**: [_________]
- **local minimum value**: [_________]
By conducting these steps, one can effectively determine the local maximum and minimum values of \( f(x) \) using mathematical analysis. This method provides a systematic approach to understanding the behavior of polynomial functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6653f5ea-9ae4-4965-89e5-4546e31fc625%2F51577421-67b3-4b84-a614-7cacb90e4f37%2Ftt0jzr.png&w=3840&q=75)
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