1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
larisa [96]
2 years ago
11

I need help with problem 1 with a through explanation and solution please ​

Mathematics
1 answer:
ki77a [65]2 years ago
6 0

Explanation:

The cubic ...

  f(x) = ax³ +bx² +cx +d

has derivatives ...

  f'(x) = 3ax² +2bx +c

  f''(x) = 6ax +2b

<h3>a)</h3>

By definition, there will be a point of inflection where the second derivative is zero (changes sign). The second derivative is a linear equation in x, so can only have one zero. Since it is given that a≠0, we are assured that the line described by f''(x) will cross the x-axis at ...

  f''(x) = 0 = 6ax +2b   ⇒   x = -b/(3a)

The single point of inflection is at x = -b/(3a).

__

<h3>b)</h3>

The cubic will have a local extreme where the first derivative is zero and the second derivative is not zero. These will only occur when the discriminant of the first derivative quadratic is positive. Their location can be found by applying the quadratic formula to the first derivative.

  x=\dfrac{-2b\pm\sqrt{(2b)^2-4(3a)(c)}}{2(3a)} = \dfrac{-2b\pm\sqrt{4b^2-12ac}}{6a}\\\\x=\dfrac{-b\pm\sqrt{b^2-3ac}}{3a}\qquad\text{extreme point locations when $b^2>3ac$}

There will be zero or two local extremes. A local extreme cannot occur at the point of inflection, which is where the formula would tell you it is when there is only one.

__

<h3>c)</h3>

Part A tells you the point of inflection is at x= -b/(3a).

Part B tells you the midpoint of the local extremes is x = -b/(3a). (This is half the sum of the x-values of the extreme points.) You will notice these are the same point.

The extreme points are located symmetrically about their midpoint, so are located symmetrically about the point of inflection.

_____

Additional comment

There are other interesting features of cubics with two local extremes. The points where the horizontal tangents meet the graph, together with the point of inflection, have equally-spaced x-coordinates. The point of inflection is the midpoint, both horizontally and vertically, between the local extreme points.

You might be interested in
For charles the great please help
dmitriy555 [2]

Answer:

119

Step-by-step explanation:

23 plus 64 plus 32

I think I just did this question

8 0
2 years ago
The 20 tallest buildings in the world have an average height of 1,667.25 feet, with a standard deviation of 318.31 feet. The Zif
valentina_108 [34]

Answer:

Marina, Wuhan center, Zifeng Tower , Shangai Tower

Step-by-step explanation:

Using the z-scores of the Shangai tower and the Marina, we can calculate their heights

Mathematically, the z-score or standard score can be calculated as follows

Zscore =  X - μ/σ

where σ and μ represents the standard deviation and the population mean respectively.

For Shaghai Tower ;

1.27 = (X - 1667.25)/318.31

318.31 × 1.27 = X - 1667.25

404.25 = X - 1667.25

X = 1667.25 + 404.25 = 2071.50

The height of the Shaghai Tower is 2071.50 feet

For Marina, we use the same procedure;

-0.84 = (X - 1667.25)/318.31

318.31 × -0.84 = X - 1667.25

-267.38 = X - 1667.25

X = 1667.25 - 267.38 = 1399.87

The height of Marina building is 1399.87

Listing the buildings' height in ascending order we have; Marina, Wuhan center, Zifeng Tower , Shangai Tower

The reasoning behind this can be mathematically explained in terms of the z-score. The standard score of the heights of the building give us an insight as to how tall the buildings can be.

Since they are in the same population i.e building heights, the higher the z-score, the higher the height of the building. A positive z-score indicates a height taller than the average while a negative z-score indicate a height lower than the average height which is the mean

6 0
3 years ago
Which choice best describes the angles st
Aleks04 [339]
Supplementary because the add to 180°
8 0
2 years ago
A different way of proving the angle sum of a triangle is to use diagram
marishachu [46]

a.yes

Step-by-step explanation:

a.angle BCD =b because angle BCD is alternate angle of angle ABC

b.the two angles angle BAC and angle ACD is co-interior angle because they are inside a parallel line

7 0
3 years ago
PLEASSE HEP MEEEEEEEE
DENIUS [597]

Answer:

s = 70

Step-by-step explanation:

\frac{s}{10}  = 7

10 \times  \frac{s}{10}  = 10 \times 7

s = 70

<h2>Insta: 25k_kem</h2>
3 0
2 years ago
Other questions:
  • Write 0.4% as a fraction explain
    15·2 answers
  • A variable that cannot be measured in terms of how much or how many but instead is assigned values to represent categories is ca
    8·2 answers
  • If Company X has 1600 employees and 80% of those employees have attended the warehouse training course how many employees have y
    12·1 answer
  • Shirley wants to buy a skateboard for $64. She has $98 in her account. She spent $10.85 to buy stationary. She also wants to buy
    10·2 answers
  • A bag is filled with 4 red marbles, 3 blue marbles, 3 yellow marbles, and
    5·1 answer
  • What is the median ??
    12·2 answers
  • 500% of what number is 3,700?
    7·1 answer
  • Question 11 (1 point)<br> John received 21 out of 30 on his test. What percentage would that be?
    8·2 answers
  • Help pls- And now brainly is making me type more soo-​
    12·1 answer
  • "I know the tree, but what is the root that produced it?"In this case the tree is "216", what is it's cube root?
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!