Aubrey and Charlie are driving to a city that is 120 mi from their house. They have already traveled 20 mi, and they are driving at a constant rate of 50 mi/h. Complete the function that models the distance they drive as a function of time. Then complete a reasonable domain for this situation. function notation in slope-intercept form: f(x) = reasonable domain: 0 SXS
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.
![### Problem Statement
Aubrey and Charlie are driving to a city that is 120 miles from their house. They have already traveled 20 miles, and they are driving at a constant rate of 50 miles per hour. Complete the function that models the distance they drive as a function of time. Then complete a reasonable domain for this situation.
### Task
**Function Notation in Slope-Intercept Form:**
\[ f(x) = \]
**Reasonable Domain:**
\[ 0 \leq x \leq 2 \]
### Explanation
- **Slope-Intercept Form:** The problem provides the rate at which Aubrey and Charlie are driving (50 miles per hour) and the initial amount of distance already traveled (20 miles). Use this information to write the function for total distance in the form of \( f(x) = mx + b \), where \( m \) represents the rate of travel and \( b \) represents the initial distance traveled.
- **Reasonable Domain:** The domain is the range of values that are valid for \( x \) in the context of this problem. Since they are traveling up to a certain point, consider the time \( x \) in hours that it will take them to reach the city.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4db8a38b-169d-4d89-a1dc-4a0b6a79633d%2Ff3d7a765-456d-4db2-b59c-04b96c8696d8%2Fmmidse_processed.png&w=3840&q=75)
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