Fixed Point of a function

In summary, a fixed point of a function is a value at which the input and output of the function are equal. To find the fixed point, you can either graph the function and look for the point where the line intersects with the line y=x, or algebraically solve for the point where the input and output are equal. The fixed point of a function is important because it can give insight into the behavior and properties of the function, and can be used to analyze the stability or convergence of a system. A function can have multiple fixed points, and some functions may have an infinite number of fixed points. A fixed point is different from a stationary point, as a stationary point is a point on a graph where the slope is zero. Not
  • #1
yayMath
6
0
Hi,
I need to prove that f(x) has a fixed point, given that f'(x) >= 2 for all x.

my problem is that I've reached the part in which g(x) = f(x) - x
and g'(x) = f'(x) - 1 and therefore g'(x) >= 1
but now I'm completely stuck. I knwo that I need to use the mean value theorem, but i just don't know where.
thanx.
 
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  • #2
If f '(x) >= 2 for all x then its graph would have to cross the graph of y = x at some point.
 
  • #3


Firstly, let's define what a fixed point of a function is. A fixed point of a function f(x) is a value x0 such that f(x0) = x0. In other words, when we plug in x0 into the function, the output is equal to the input.

Now, let's consider the function g(x) = f(x) - x. We know that g'(x) = f'(x) - 1 and since we are given that f'(x) >= 2 for all x, we can conclude that g'(x) >= 1 for all x. This means that g(x) is a strictly increasing function.

Next, we can use the mean value theorem to show that g(x) has a fixed point. The mean value theorem states that for a continuous and differentiable function f(x) on an interval [a,b], there exists a point c in the interval such that f'(c) = (f(b) - f(a))/(b-a).

In our case, since g(x) is a strictly increasing function, it is also continuous and differentiable on any interval. Therefore, we can apply the mean value theorem to g(x) on any interval [a,b]. This means that there exists a point c in the interval [a,b] such that g'(c) = (g(b) - g(a))/(b-a).

Since we know that g'(x) >= 1 for all x, we can say that g(b) - g(a) >= b-a for any interval [a,b]. This means that g(b) >= g(a) + b-a.

Now, let's consider the interval [a,a+1]. Since g(a+1) >= g(a) + (a+1)-a = g(a) + 1, we can say that g(a+1) >= g(a) + 1.

Similarly, we can consider the interval [a+1,a+2] and apply the same logic to get g(a+2) >= g(a+1) + 1.

Continuing this process, we can see that g(a+n) >= g(a+n-1) + 1 for all n > 0.

Now, let's consider the sequence {g(a+n)}. This sequence is strictly increasing and since g(x) is continuous, we know that g(a+n) approaches a limit
 

What is a fixed point of a function?

A fixed point of a function is a value at which the input and output of the function are equal. In other words, when the input is substituted into the function, the output remains the same.

How do you find the fixed point of a function?

To find the fixed point of a function, you can either graph the function and look for the point where the line intersects with the line y=x, or algebraically solve for the point where the input and output are equal.

Why is the fixed point of a function important?

The fixed point of a function is important because it can give insight into the behavior and properties of the function. It can also be used to analyze the stability or convergence of a system.

Can a function have more than one fixed point?

Yes, a function can have multiple fixed points. In fact, some functions may have an infinite number of fixed points.

What is the difference between a fixed point and a stationary point of a function?

A fixed point is a value at which the input and output of a function are equal, while a stationary point is a point on a graph where the slope is zero. Not all stationary points are fixed points, but all fixed points are stationary points.

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