For each of the following functions, determine all horizontal intercepts. Try to determine these algebraically first then verify your answers by graphing the functions on your calculator. а. h (ӕ) — 4(а — 8) + 2 Preview b. b(ӕ) — г(12 — 2г) (9 — 2г) - Preview
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.
![The actual distance (in miles) is 28 times as large as the corresponding distance on the map (in inches). This table shows pairs of lengths on the map and the corresponding actual distances. The graph on the right shows the same pairs represented in the table as points and the trace that shows all possible pairs of corresponding distances on the map (in inches) and actual distances (in miles).
| distance on the map | actual distance |
|---------------------|-----------------------|
| 1.25 inches | 28(1.25) = 35 miles |
| 2.7 inches | 28(2.7) = 75.6 miles |
| 6.5 inches | 28(6.5) = 182 miles |
| 14 inches | 28(14) = 392 miles |
### Graph Description
The graph is a line plot showing "actual distance (in miles)" on the vertical axis and "distance on the map (in inches)" on the horizontal axis. The line passes through the points:
- (1.25, 35)
- (2.7, 75.6)
- (6.5, 182)
- (14, 392)
The graph clearly illustrates the linear relationship between the distances.
We can scan the above graph and see that each value of the distance in inches on the map, \( n \), (indicated on the horizontal axis) corresponds to exactly one distance (in miles) on the road, \( d \).
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For each of the following functions, determine all horizontal intercepts. *Try to determine these algebraically first, then verify your answers by graphing the functions on your calculator*:
a. \( h(x) = 4(x - 8) + 2 \)
\[ x = \quad \text{Preview} \]
b. \( b(x) = x(12 - 2x)(9 - 2x) \)
\[ x = \quad \text{Preview} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2810685e-1eeb-461a-9149-0ef27d84930f%2Fd5eea916-ea38-41e0-b85a-b204673ea331%2Fk8j2nta_processed.png&w=3840&q=75)
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