What does it mean when critical points have non-real answers?

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In summary, the person is asking for help in finding the critical X's and Local Max/Min points for the equation y = x/x^2-1. They mention that they know they need to take the derivative and equate it to zero to find the critical X's, but they got a non-real answer for one of the critical points. They also mention finding the critical X's when y is undefined to find the asymptotes. They suggest going to the website www.calc101.com to see that there are no points where the gradient is actually zero, but it approaches zero at plus and minus infinities. Other critical points include where the graph's derivative goes to infinity, at plus and minus 1.
  • #1
Haroldoo
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Hi, I was wondering if I could get an answer to a question that has been bothering me all day today.

Ok, so I got an assignment today and it wants me to find the Critical X's and Local Max/Min points for the equation: y = x/x^2-1

Correct me if I'm wrong on any of this. To find the critical X's I take the derivative and equate it to zero then solve for x to get the critical X's and from there I can find the Max/Min vaules.

After doing that I got x=sqrt(-1). Now I know that is a non real answer, so does that mean that those critical points do not exist? and does that also mean that there is no local max/min points? What exactly does it mean when I get the non real answers for my critical x values. Oh and I also found the critical x's when y is undefined to find the asymptotes.

If i wrote this like crap and you can't understand it, sorry heh, but any feedback is appreciated
 
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If you were to go to www.calc101.com and go to the graphing section, input your function and you will see that there are no points where the gradient is actually zero, although it does approach zero at plus and minus infinities. Other critical points include where the graph's derivative goes to infinity, at plus and minus 1.
 
  • #3


Hi there,

When critical points have non-real answers, it means that those points do not exist on the real number line. In your case, finding x=sqrt(-1) as a critical point means that there is no critical point for the given equation. This also means that there are no local max/min points for the equation.

When taking the derivative and setting it equal to zero, you are finding the points where the slope of the function is zero, which can indicate a maximum or minimum point. However, in this case, the derivative does not have any real solutions, so there are no critical points.

It is important to note that just because there are no critical points, it does not mean that the function does not have any extrema (maximum or minimum points). It is possible for a function to have extrema without any critical points.

I hope this helps clarify the concept for you. Let me know if you have any other questions.
 

FAQ: What does it mean when critical points have non-real answers?

What is meant by "curve sketching problem"?

The curve sketching problem is a mathematical process of analyzing a function and creating a rough sketch of its graph. This involves determining the behavior of the function, finding its critical points, and identifying the shape of the graph.

Why is curve sketching important in mathematics?

Curve sketching is important in mathematics because it allows us to visualize and understand the behavior of a function. It also helps in solving real-world problems, such as finding maximum or minimum values, optimizing functions, and understanding the relationship between different variables.

What are the steps involved in curve sketching?

The steps involved in curve sketching include finding the domain and range of the function, determining the behavior of the function at the edges of its domain, finding the critical points and their nature, analyzing the concavity and inflection points, and plotting the points on a graph to create a rough sketch.

How do you determine the concavity of a function?

To determine the concavity of a function, you need to find the second derivative of the function and evaluate it at different points in the domain. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

What are some common mistakes to avoid in curve sketching?

Some common mistakes to avoid in curve sketching include forgetting to check for asymptotes, incorrectly identifying critical points, not considering the behavior of the function at the edges of its domain, and not plotting enough points to accurately represent the graph.

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