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
Anastasy [175]
3 years ago
12

a) Suppose f"(z) exists on an interval I and f(s) has a zero at three distinct points a < b< c on I. Show there is a pbint

p on [a, c] where f"(p) = 0. b)mustrate part (a)on the cubic f(s) = (x- a)(x - b)(x - c)
Mathematics
1 answer:
Fynjy0 [20]3 years ago
3 0

Answer:

We can find a root in the average point (a+b+c)/3

Step-By-Step Explanation:

We can use the Rolle Theorem. Since f is two times deribable on both [a,b] and [b,c], then there exists points x in (a,b) and y in (b,c) such that f'(x) = f'(y) = 0. Now, again by using Rolle Theorem, since f' is derivable in [x,y] (because it is a closed interval in I), then there exists s in [x,y] such that f''(s) = 0. This proves a.

The cube (x-a)(x-b)(x-c) has 3 roots, a, b and c and it is two times derivable because it is a polynomial. Hence we can use (a) to ensure that there is a root on I. Nevertheless, we can try to find the root manually:

f'(x) = (x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)

f''(x) = (x-c)+(x-b)+(x-c)+(x-a)+(x-b)+(x-a) = 2(x-a)+2(x-b)+2(x-c) = 2( (x-a) + (x-b) + (x-c) ) = 6x - 2 (a+b+c).

We want x such that

6x - 2(a+b+c) = 0, or, equivalently,

3x - (a+b+c) = 0

Hence

x = (a+b+c)/3

Is a root of f'' in I.

You might be interested in
Lucy is saving money to buy a game. The game costs $21, and so far she has saved one-third of this cost. How much money has Lucy
Svetach [21]

Answer:

$7

Step-by-step explanation:

just did 21 divided by three since it says she has saved one third of the $21.

5 0
2 years ago
Read 2 more answers
Mandy drew a scale drawing of a summer camp. The scale of the drawing was 3 centimeters : 2 meters. The sand volleyball court is
stellarik [79]

Answer:

18 m

Step-by-step explanation:

27/3 = 9

9 x 2 = 18 m

5 0
3 years ago
Select the procedure that can be used to show the converse of the Pythagorean theorem using side lengths chosen from 6 cm, 8 cm,
Effectus [21]

Answer: The answer is B have A great day

Step-by-step explanation:

3 0
2 years ago
Read 2 more answers
What is the circumference? Use 3.14 for π. Round to the nearest tenth if necessary.
Phantasy [73]

Answer:

I think it's A. 12.1 mm

<h2 />

6 0
2 years ago
Read 2 more answers
What day and time was it 2,020 minutes after the beginning of January 1, 2020
Rudiy27

Answer:

January 2, 2020

Time: 2:20pm

4 0
2 years ago
Other questions:
  • Luci cuts a board that is 3/8 yard long into pieces that are 3/16 yard long. How many pieces does she cut?
    12·1 answer
  • What rule relates the number of circles in the pattern, c, to the figure number, f ?
    5·1 answer
  • What is the slope of y= -3x -2
    7·1 answer
  • Dominic bought a new alarm car that was on sale for $18.35 if this price represents a 30% discount from the original price what
    10·1 answer
  • The annual Gross Domestic Product (GDP) of a country is the value of all of the goods and services produced in the country durin
    9·1 answer
  • How do this pleassssse
    5·1 answer
  • Tickets to the haunted house are $6.50 for adults and $4.30 dor childeren. How much would it cause for two adults and four child
    12·2 answers
  • Lines that are perpendicular have ___________________ slopes.<br>A. opposite reciprocal<br>b. same
    13·1 answer
  • Can someone help with 3 and 9
    15·1 answer
  • Write the explicit formula for an, the nth term of the sequence 38, 48, 58
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!