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Quadratic Equation
When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.
Demand and Supply Function
The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
![Problem 5 (Bonus: Banach fixed point theorem). Let f be a continuous
function such that
\f (x) – f(y)| < a\x – y|
(о.1)
for a constant a satisfying 0 < a < 1. The purpose of this exercise is to prove
that there is an x such that f(x)
point theorem.
[Comment. While this theorem does not seem directly related to differen-
= x. This is known as the Banach fixed
tial equations, the method of its proof is used in the proof of the existence
and uniqueness theorem for differential equations. You can read more about
this in Boyce-DiPrima section 2.8 and Braun section 1.10.]
Write a complete proof of the theorem by filling the gaps in these steps:
1. Choose any xo and define a sequence xn inductively by
Xn = f(xn-1).
That is: x1 = f(x0), x2 = f (x1), and so on.
2. Using (o.1), show that for every n
|Xn – Xp-1|< a"-1|x1 – xol-
3. Write x, as
Xn =
(Xn – Xn-1) + (Xn–1 – Xn–2) +.. +(x1 – xo) + xo.
4. By comparing the above sum to the convergent series Eo a" conclude
that the sequence xn converges as n → ∞,
5. Let x = lim,00 Xn. Justify that we can pass to the limit n → 0 in
Xn = f(xn-1)
and conclude that x satisfies f(x)
6. Using (0.1), show that if y is any number satisfying f(y) = y, then
y = x, that is: there is only one solution to the equation f (x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05d25ab8-1881-42ba-9f18-580f5b6b87d7%2F00776d79-bd65-4c50-ada3-9b9efc3f72f5%2Fz6ilvnx_processed.png&w=3840&q=75)
We the continuity of f to show that f has unique fixed point
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