Critical Point
Definition
A critical point is a point on a function where the derivative is either zero or undefined. It represents a potential maximum, minimum, or inflection point.
Analogy
Think of a critical point as a crossroad in your journey. At this point, you can either take a different path (representing a maximum or minimum) or continue straight ahead (representing an inflection point).
Related terms
Local Maximum: A local maximum is a critical point where the function reaches its highest value within a small interval around that point.
Local Minimum: A local minimum is a critical point where the function reaches its lowest value within a small interval around that point.
Inflection Point: An inflection point is a critical point where the concavity of the function changes from upward to downward or vice versa.
"Critical Point" appears in:
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Study guides (3)
Practice Questions (14)
What does it mean for a function to have a critical point?
If the derivative of a function is positive on the left side of a critical point and negative on its right side, what can we conclude about the critical point?
When optimizing a function, what does it mean to find a critical point?
How can you determine whether a critical point corresponds to a maximum or minimum value in an optimization problem?
What does it indicate if the derivative is negative on one side of a critical point and positive on the other side?
What does it indicate if the derivative is positive on both sides of a critical point?
What does it indicate if the derivative is negative on both sides of a critical point?
What does it mean when the y-value of a critical point is larger than the y-values of the endpoints?
What does it mean when the y-value of a critical point is smaller than the y-values of the endpoints?
If the second derivative is negative at a critical point, then the critical point corresponds to:
If the second derivative is positive at a critical point, then the critical point corresponds to:
Which of the following statements is true about a function if its second derivative is zero at a critical point?
The Second Derivative Test is used to analyze the concavity of a function and determine whether a critical point is a relative minimum, relative maximum, or neither. This test is based on the principle that:
What key feature of a function can be determined by analyzing the sign of the first derivative at a critical point?
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