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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### Finding Intervals of Positivity for a Given Function
**Problem Statement:**
You are given the function:
\[ f(x) = -2(x + 3.5)(x - 0.5)(x - 3) \]
Your task is to find the intervals on which \( f(x) \) is positive by plotting a graph.
#### Steps to Determine Intervals of Positivity:
1. **Identify Critical Points:**
- Find the roots of the function where \( f(x) = 0 \).
For the given function:
\[
x + 3.5 = 0 \implies x = -3.5
\]
\[
x - 0.5 = 0 \implies x = 0.5
\]
\[
x - 3 = 0 \implies x = 3
\]
The critical points are \( x = -3.5 \), \( x = 0.5 \), and \( x = 3 \).
2. **Plot the Graph:**
- Sketch the polynomial function on a graph considering the critical points.
- Determine where the function crosses the x-axis: at \( x = -3.5 \), \( x = 0.5 \), and \( x = 3 \).
- The function \( f(x) \) is negative at the immediate neighborhood of each root between two adjacent points.
3. **Determine Intervals:**
- Analyze the sign of the function in each interval created by the critical points:
- Check intervals: \( (-\infty, -3.5) \), \( (-3.5, 0.5) \), \( (0.5, 3) \), and \( (3, \infty) \).
- Test a point in each interval to see if \( f(x) \) is positive or negative.
4. **Conclusion:**
- The intervals where \( f(x) \) is positive will be the ranges between the critical points where the function value is greater than zero.
**Final Answer:**
\[ f(x) \text{ is positive on } \big((-\infty, -3.5) \cup (0.5, 3)\big). \]
**Note:**
This is a general step-by-step method; to fully confirm](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84855d5a-1164-4701-aa00-ab1e8c4fccea%2F21b76cf9-19d7-4a7a-8ed1-34c71bae6e2c%2Fea6wux6_processed.jpeg&w=3840&q=75)
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