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
OverLord2011 [107]
3 years ago
10

Let f be a continuous function defined on the interval [0,1] such that f(0) = f(1). Show that there exists a number 0 <= a &l

t;= 1/2 such that f(a) = f(a+1/2)
Mathematics
1 answer:
Leni [432]3 years ago
5 0

Explanation:

In order to prove that affirmation, we define the function g over the interval [0, 1/2] with the formula g(x) = f(x+1/2)-f(x) .

If we evaluate g at the  endpoints we have

g(0) = f(1/2)-f(0) = f(1/2) - f(1) (because f(0) = f(1))  

g(1/2) = f(1) - f(1/2) = -g(0)

Since g(1/2) = -g(0), we have one chance out of three

  • g(0) > 0 and g(1/2) < 0
  • g(0) < 0 and g(1/2) > 0
  • g(0) = g(1/2) = 0

We will prove that g has a zero on [0,1/2]. If g(0) = 0, then it is trivial. If g(0) ≠ 0, then we are in one of the first two cases, and therefore g(0) * g(1/2) < 0. Since f is continuous, so is g. Bolzano's Theorem assures that there exists c in (0,1/2) such that g(c) = 0.  This proves that g has at least one zero on [0,1/2].

Let c be a 0 of g, then we have

0 = g(c) = f(c+1/2)-f(c)

Hence, f(c+1/2) = f(c) as we wanted.

You might be interested in
A man and his three children spent $40 to attend a show. A second family of three children and their two parents spent $53 for t
wariber [46]
The cost for the children tickets is $9
5 0
2 years ago
Find the LCM of the following
Tresset [83]

Answer:

A. LCM of 6,9, and 15 is 90.

B. LCM of 3,12, and 16 is 48.

C. LCM of 8,9, and 12 is 72.

D. LCM of 24 and 18 is 72.

E. LCM of 21 and 28 is 84.

F. LCM of 19 and 42 is 798.

Step-by-step explanation:

Hope this helps! :)

8 0
2 years ago
A line passes through the points (7, 10) and (7, 20). Which statement is true about the line?
saul85 [17]
It has no slope 
a vertical line 
x=7


4 0
3 years ago
Read 2 more answers
The boeing 747 airplane is 250 feet in length. If a model of this plane is 8 inches long , what scale was used to create this mo
cluponka [151]

Answer: The wingspan is 36 m.

Using a proportion to solve this, we write the scale factor as a ratio first:  19/38, since 19 is the size of the model's length and 38 is the real length.  For the second ratio, we have 18 for the size of the model wingspan and x for the real wingspan:

19/38 = 18/x

Cross multiply:

19*x = 38*18

19x = 684

8 0
3 years ago
Suppose y varies inversely with x, and y = 3 when x = 1/3. What is the inverse variation equation that relates x and y ?
Sav [38]
Good the answer is going to be 1/3x3y
8 0
3 years ago
Read 2 more answers
Other questions:
  • Create a set of data that contains 11 test scores and satisfies each condition below:
    13·2 answers
  • The polynomial x3 + 4x2 – 9x – 36 has 4 terms. Use factoring by grouping to find the correct factorization.
    7·2 answers
  • I dont have a question anymmore lol k thanks
    12·1 answer
  • Help please!!!
    14·2 answers
  • HELP I WILL MARK YOU AS A BRAINLIEST!!
    9·1 answer
  • Find the slope of the line passing through (-9,-6) and (-4,5)
    11·2 answers
  • Element X is a radioactive isotope such that its mass decreases by 73% every
    12·1 answer
  • Hi! please help i’ll give brainliest
    10·2 answers
  • Solve for x , y , right answers plss !!!!
    14·1 answer
  • Area of a triangle when b = 6 m and h=7 m
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!