(b) Find the vertex of the graph. (х, Р) %3D Is it a maximum point or a minimum point? maximum minimum (c) Is the average rate of change of this function from x = a < 125 to x = 125 positive or negative? positive negative (d) Is the average rate of change of this function from x = 125 to x = a > 125 positive or negative? positive negative (e) Does the average rate of change of the profit get closer to or farther from 0 when a is closer to 125? closer to 0 farther from 0
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.
![**Vertex and Rate of Change Questions**
(b) Find the vertex of the graph.
\[
(x, P) = \left( \underline{\hspace{1cm}} \right)
\]
Is it a maximum point or a minimum point?
- ☐ maximum
- ☐ minimum
(c) Is the average rate of change of this function from \( x = a < 125 \) to \( x = 125 \) positive or negative?
- ☐ positive
- ☐ negative
(d) Is the average rate of change of this function from \( x = 125 \) to \( x = a > 125 \) positive or negative?
- ☐ positive
- ☐ negative
(e) Does the average rate of change of the profit get closer to or farther from 0 when \( a \) is closer to 125?
- ☐ closer to 0
- ☐ farther from 0
**Explanation:**
This section focuses on identifying the vertex of a given graph and analyzing the average rate of change around a particular point, \( x = 125 \). It involves determining whether the vertex is a point of maximum or minimum value and evaluating how the rate of change behaves as the variable approaches or moves away from a specific point on the graph. The questions help students understand crucial concepts in calculus related to functions and their rates of change, using critical thinking to determine positivity or negativity in different intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2ed13a8-94cc-42b1-92cb-21374826aa79%2F66fea626-8a6a-404d-bab2-f29288d93bbf%2Fdavzgbf_processed.png&w=3840&q=75)
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