....... ...................... ...................... ......... The graph of a function f is given. Use the graph to estimate the following. (Enter your answers using interval notation.) y WebAssign Plot (a) The domain and range of f. domain range (b) The intervals on which f is increasing and on which f is decreasing. increasing decreasing
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.
- **Axes**: The \( x \)-axis and \( y \)-axis are both marked.
- **Plot**: The function \( f \) is plotted as a red curve.
- **Intervals**: The graph shows parts where the curve increases and decreases.
#### (a) Domain and Range of \( f \)
- **Domain**: The set of all possible values of \( x \) for which the function \( f \) is defined.
- **Range**: The set of all possible values of \( y \) that the function \( f \) can take.
| Domain | \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ |
| Range | \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ |
#### (b) Intervals of Increasing and Decreasing \( f \)
- **Increasing**: The set of intervals where the function \( f \) rises as \( x \) increases.
- **Decreasing**: The set of intervals where the function \( f \) falls as \( x \) increases.
| Increasing | \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ |
| Decreasing | \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ |
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**Explanation of the Graph:**
The graph shows a function with the following characteristics:
- **Intervals of Increase**: Identify where the curve ascends as you move from left to right.
- **Intervals of Decrease**: Identify where the curve descends as you move from left to right.
- **Maximum and Minimum Points**: Locations where the function changes direction from increasing to decreasing (and vice versa).
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Use this information to fill in the blanks for the domain, range, and intervals where the function is increasing and decreasing. Make sure to use interval notation to express your answers accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F429fd668-1075-4973-8e47-64e8dad266df%2Fcb1deb6b-2b26-48f2-a1a4-400b1d3e8d22%2Fkw4msj9_processed.png&w=3840&q=75)
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