Identifying and Classifying Critical Points

In summary, critical points are points on a graph where the derivative is equal to zero, indicating a possible maximum, minimum, or inflection point. They are important in science and can be determined using the second derivative test. A function can have multiple critical points, and they are crucial in optimization problems as they help us find the optimal solution. However, critical points may not always be visible on a graph and may require calculus techniques for analysis.
  • #1
jegues
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Homework Statement



Find all critical points of the function given.(see figure)

Choose and classify anyone critical point.

Homework Equations





The Attempt at a Solution



I found the following critical points,

(0,0), (0,-3), (3/2,3/2) and (-3/2, 3/2)

Wolfram alpha is telling me the critical points are (0,0) and (0,3). (We don't deal with critical points that have complex numbers in them).

I can't find my error.

Can anyone see it?
 

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  • #2
You flipped a sign when you factored the y terms in equation (1).
 

FAQ: Identifying and Classifying Critical Points

What are critical points and why are they important in science?

Critical points are points on a graph where the derivative is equal to zero, indicating a possible maximum, minimum, or inflection point. They are important in science because they help us understand the behavior and changes of a system, such as in physics, chemistry, or economics.

How do you determine if a critical point is a maximum, minimum, or inflection point?

To determine the type of critical point, we can use the second derivative test. If the second derivative is positive, the point is a minimum. If the second derivative is negative, the point is a maximum. If the second derivative is zero, the point could be either an inflection point or a critical point that requires further analysis.

Can a function have more than one critical point?

Yes, a function can have multiple critical points. For example, a cubic function can have up to three critical points. These points can be either maximum, minimum, or inflection points.

How do critical points relate to optimization problems?

Critical points are important in optimization problems as they help us find the maximum or minimum value of a function. By finding the critical points and using the first or second derivative test, we can determine the optimal solution for a given problem.

Are critical points always visible on a graph?

No, critical points are not always visible on a graph. Sometimes, they can be located at points where the graph changes direction or curvature, but other times they may be hidden within the graph. In these cases, it is important to use calculus techniques to find the critical points and analyze their behavior.

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