# Critical points (will be a function of beta and delta)

• Rubik
In summary, to find critical points for this system of equations, we need to solve them simultaneously and check the solutions. The values of β and δ determine the existence of critical points through the discriminant of the resulting quadratic equation.
Rubik

## Homework Statement

Find the critical points where $\alpha$ = 1.

dv/dt = v2 + $\alpha$v - u + $\delta$ [I will call this (1)]

du/dt = $\beta$v - u [call this (2)]

For what values of $\beta$ and $\delta$ are there no critical points?

## The Attempt at a Solution

So firstly I set (1) and (2) to = 0 and from (2) I get:
u = $\beta$v and now I subst. this back into (1)

0 = v2 + v(1 - $\beta$) + $\delta$

Now from the quadratic formula I have this

v = {-(1 - $\beta$) ± √[(1 - $\beta$)2 - 4$\delta$]}/2

Now here I am stuck and not sure how to simplify this or interpret this?

I would like to start by clarifying that the problem statement is not complete. It is missing important information such as the initial conditions and the domain of the variables. Without this information, it is difficult to provide a complete and accurate solution.

That being said, I will try to provide some general guidance on finding critical points for this system of equations.

To find the critical points, we need to solve the system of equations (1) and (2) simultaneously. This means that we need to find values of v and u that satisfy both equations at the same time. One way to approach this is to substitute the expression for u from equation (2) into equation (1), as you have done. This will give you a quadratic equation in terms of v.

Once you have this quadratic equation, you can use the quadratic formula to solve for v. This will give you two possible values for v, which we can call v1 and v2. These values represent the potential critical points.

To determine which of these values are actually critical points, we need to substitute them back into both equations (1) and (2) and see if they satisfy both equations. If they do, then they are critical points. If they do not, then they are not critical points.

Now, to answer the question about the values of β and δ for which there are no critical points, we need to consider the discriminant of the quadratic equation we obtained earlier. The discriminant is the part inside the square root in the quadratic formula, and it determines the nature of the solutions to the equation. If the discriminant is negative, then there are no real solutions, which means there are no critical points in this case.

So, to summarize, to find the critical points for this system of equations, we need to solve the equations simultaneously and check if the solutions satisfy both equations. And to determine the values of β and δ for which there are no critical points, we need to consider the discriminant of the quadratic equation obtained from the first step.

I hope this helps you in finding the critical points and understanding the role of β and δ in determining the existence of critical points. Remember to always provide all necessary information in a problem statement to ensure a complete and accurate solution.

## 1. What are critical points?

Critical points are points on a graph where the derivative (slope) of a function is equal to zero. They are also known as stationary points because the function stops changing at these points.

## 2. How do critical points vary with beta and delta?

The location of critical points is influenced by the values of beta and delta. As these values change, the critical points on the graph will also shift. This is because beta and delta affect the shape and behavior of the function.

## 3. How do I find critical points?

To find critical points, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to find the x-values of the critical points. You can also use a graphing calculator or software to plot the function and identify the critical points visually.

## 4. Why are critical points important?

Critical points are important because they help us understand the behavior of a function. They can tell us when a function is increasing, decreasing, or has a maximum or minimum value. Critical points are also used in optimization problems to find the most efficient solution.

## 5. Can a function have multiple critical points?

Yes, a function can have multiple critical points. This can happen when the derivative of the function is equal to zero at more than one point. These points can be local (maximum or minimum) or inflection points. The number of critical points can vary depending on the complexity of the function and the values of beta and delta.

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