Complete the description of the piecewise function graphed below. 6+ 5- 4- -7 -6 -4 -3 -2 -1 6 7 -2 -3 -4 -5 { if -6 < x < – 3 | f(x) = { if - 3 < x < 1 { if 1< x < 6
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.
![### Piecewise Function Description
To describe the piecewise function graphed below, we need to analyze each segment of the graph individually. The graph shows three distinct segments:
1. For the interval \(-6 \leq x \leq -3\)
2. For the interval \(-3 < x \leq 1\)
3. For the interval \(1 < x \leq 6\)
The graph includes lines and points to describe the behavior and values of the function \( f(x) \) within these intervals.
#### Graph Description:
- **For \( -6 \leq x \leq -3 \):**
- The graph shows a line segment starting at the point \((-6, -5)\) and ending at the point \((-3, 4)\).
- This line can be described by the linear equation connecting these points.
- **For \( -3 < x \leq 1 \):**
- The graph shows a horizontal line starting right after the point \((-3, 4)\) and ending at the point \((1, 2)\).
- A horizontal line signifies that the function value is constant within this interval.
- **For \( 1 < x \leq 6 \):**
- The graph shows a line segment starting right after the point \((1, 2)\) and extending to the point \((6, -6)\).
- This line can also be described by the linear equation connecting these points.
The complete piecewise function can thus be defined as follows:
\[ f(x) =
\begin{cases}
\frac{3x+13}{3} & \text{if } -6 \leq x \leq -3 \\
4 & \text{if } -3 < x \leq 1 \\
-\frac{8x-6}{5} & \text{if } 1 < x \leq 6
\end{cases}
\]
The equations are derived from the two-point form of a line equation for each of the linear segments identified in the graph.
#### Diagram Explanation:
The graph is on a Cartesian plane with:
- The x-axis ranging from -7 to 7.
- The y-axis ranging from -6 to 6.
- Blue lines indicating the segments of the function with solid dots signifying inclusive](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f072395-e0a4-486f-9c92-023b53ab9ed9%2F08475c60-b79c-46c0-9579-2f71dae02ecd%2Fhri43s_processed.png&w=3840&q=75)
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