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kozerog [31]
3 years ago
12

Suppose that f : [a, b] → [a, b] is continuous. Prove that f has a fixed point. That is, prove that there exists c ∈ [a, b] such

that f (c) = c.
Mathematics
1 answer:
nikklg [1K]3 years ago
6 0

Answer:

Step-by-step explanation:

define the function:

g(x) = f(x) -x

As both f(x) and x are continuous functions, g(x) will also be continuous.

Now, what can we say about g(a) = f(a) -a?

we know that a\leq f(a) \leq b, thus:

a-a\leq f(a)-a \leq b-a\\0 \leq g(a) \leq b-a

thus  g(a) is non-negative.

What about g(b) ? Again we have:

a\leq f(b) \leq b\\a-b \leq f(b) -b \leq  0\\a-b \leq g(b) \leq  0

That means that g(b) is not positive.

Now, we can imagine two cases, either one of g(a) or g(b) is equal to zero, or none of them is. If either of them is equal to zero, we have found a fixed point! In fact, any point c for which g(c)=0 is a fixed point, because:

g(c) = 0 \implies f(c) -c = 0 \implies f(c) = c

Now, if g(a) \neq  0 and g(b) \neq 0, then we have that

g(a) >0 and g(b) < 0. And by Bolzano's theorem we can assert that there must exist a point c between a and b for which g(c)=0. And as we have shown before that point would be a fixed point. This completes the proof.

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svet-max [94.6K]
Simplify, simplify, simplify for problems like this. Add 6x to both sides to get 10x - 8 = 12. Then add 8 to both sides, so 10x = 20. Now divide by 10, and you get x = 2
5 0
3 years ago
Mustafa, Heloise, and Gia have written more than a combined total of 222222 articles for the school newspaper. Heloise has writt
baherus [9]

Answer:

The Inequality For determining number of equation written by Mustafa for school paper is x+\frac{1}{4}x+ \frac{3}{2}x\geq 22.

Mustafa has written more than 8 articles.

Step-by-step explanation:

Given:

Combined Total Number of articles = 22

Let the number of articles written by Mustafa be 'x'.

Now Given:

Heloise has written \frac{1}{4} as many articles as Mustafa has.

Number of article written by Heloise = \frac{1}{4}x

Gia has written \frac{3}{2} as many articles as Mustafa has.

Number of article written by Gia = \frac{3}{2}x

Now we know that;

The sum of number of articles written by Mustafa and Number of article written by Heloise and Number of article written by Gia is greater than or equal to Combined Total Number of articles.

framing in equation form we get;

x+\frac{1}{4}x+ \frac{3}{2}x\geq 22

Hence the Inequality For determining number of equation written by Mustafa for school paper is x+\frac{1}{4}x+ \frac{3}{2}x\geq 22.

Now Solving the Inequality we get;

Taking LCM for making the denominator common we get:

\frac{x\times 4}{4}+\frac{1\times1}{4\times1}x+ \frac{3\times2}{2\times2}x\geq 22\\\\\frac{4x}{4}+ \frac{x}{4}+\frac{6x}{4}\geq 22\\\\\frac{4x+x+6x}{4} \geq 22\\\\11x\geq 22\times4\\\\11x\geq 88\\\\x\geq \frac{88}{11} \\\\x\geq 8

Hence Mustafa has written more than 8 articles.

5 0
3 years ago
The graph shown represents the graph of an inequality. The boundary tells you that ________.
kaheart [24]
Can I have an image of the graph?
8 0
3 years ago
Find a third-degree polynomial equation with rational coefficients that has roots -4 and 6 + i.
dolphi86 [110]

Answer:

x³ - 8x² - 11x + 148

Step-by-step explanation:

Given that x = 6 + i is a root then x = 6 - i is also a root

Complex roots occur as conjugate pairs.

The factors are therefore (x - (6 + i)) and(x - (6 - i))

Given x = - 4 is a root then (x + 4) is a factor

The polynomial is the product of the factors, that is

p(x) = (x + 4)(x - (6 + i))(x - (6 - i))

      = (x + 4)(x - 6 - i)(x - 6 + i)

      = (x + 4)((x - 6)² - i²)

      = (x + 4)(x² - 12x + 36 + 1)

      = (x + 4)(x² - 12x + 37) ← distribute

      = x³ + 4x² - 12x² - 48x + 37x + 148

      = x³ - 8x² - 11x + 148

 

3 0
3 years ago
Do these pairs of values (x and y) represent two quantities that are
Alisiya [41]
B. Yes, because as x increases, y also increases

X Y
1 5 (1 x 5)
2 10 (2 x 5)
3 15 (3 x 5)
5 25 (5 x 5)
8 0
3 years ago
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