Given the function g(x) = 4x° + 6x² – 24x, find the first derivative, g'(x). = (x),6 Notice that g'’(x) = 0 when x = - 2, that is, | g'( – 2) = 0. Now, we want to know whether there is a local minimum or – 2, so we will use the second local maximum at x = derivative test. Find the second derivative, g''(x). g''(x) Evaluate g''(– 2). g''( – 2) Based on the sign of this number, does this mean the graph of g(x) is concave up or concave down at x = 2? [Answer either up or down -- watch your spelling!!] – 2 the graph of g(x) is concave At x = Based on the concavity of g(x) at x = mean that there is a local minimum or local maximum at – 2, does this - 2? X = - watch your [Answer either minimum or maximum spelling!!] -- At x = 2 there is a local |
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.
![Given the function g(x) = 4x + 6x? – 24x, find the
first derivative, g'(x).
g'(x) =
Notice that g'(x)
g'( – 2) = 0.
0 when x =
- 2, that is,
Now, we want to know whether there is a local minimum or
- 2, so we will use the second
local maximum at x
derivative test.
Find the second derivative, g''(x).
g''(x) =
Evaluate g''( – 2).
g''( – 2) =
Based on the sign of this number, does this mean the graph
- 2?
watch your spelling!!]
of g(x) is concave up or concave down at x =
[Answer either up or down
At x =
2 the graph of g(x) is concave
Based on the concavity of g(x) at x =
mean that there is a local minimum or local maximum at
- 2, does this
x =
2?
watch your
[Answer either minimum or maximum
spelling!!]
At x =
- 2 there is a local](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf7777fa-aa3c-4ddb-8a8c-67588cdbbf49%2F5c7de909-f6dd-404f-a9c6-0c561a99263e%2Foiaynu_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
