The function y = f(x) is given by the figure. y= f(x) 1 2 4 -1 -2 Find the minimum and maximum of B on [0,6]. B(x) = f(t) dt (Use decimal notation. Give your answers to two decimal places.) minimum: maximum:
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.
![**Function and Graph Analysis**
The function \( y = f(x) \) is defined by the graph shown.
The graph is a piecewise linear function that is plotted on the coordinate plane with axes labeled \( x \) and \( y \):
- The graph begins at the point (1, 0).
- It increases linearly to the point (2, 2).
- It remains constant from (2, 2) to (3, 2).
- It then decreases linearly to the point (5, 0).
- Finally, the graph continues horizontally at a value of \( y = 0 \) to the point (6, 0).
**Task**
Determine the minimum and maximum values of the function \( B(x) \) on the interval \([0, 6]\).
The function \( B(x) \) is defined as the integral:
\[
B(x) = \int_{3.5}^{x} f(t) \, dt
\]
You are required to provide the answers in decimal notation, rounded to two decimal places.
- **Minimum:**
- [Input box provided]
- **Maximum:**
- [Input box provided]
**Instructions**
- Analyze the graph carefully to understand the behavior of \( f(x) \) between different points.
- Compute the definite integral for the different segments of the graph to find \( B(x) \).
- Determine the values of \( B(x) \) at the endpoints and critical points to find the minimum and maximum values on the interval \([0, 6]\).
*Source*: Rogawski 4e Calculus Early Transcendentals | *Publisher*: W.H. Freeman](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0a9522d-164b-4f9a-91f3-55e6c1137254%2F7437b188-2d23-4078-889e-ef29f5fd5cbb%2Fm41ujql_processed.jpeg&w=3840&q=75)
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