critical numbers are numbers in the domain of the function that either make the DERIVATIVE ____ __ ____ or _____
Critical numbers are numbers in the domain of a function where either the derivative is equal to zero or the derivative does not exist
Critical numbers are numbers in the domain of a function where either the derivative is equal to zero or the derivative does not exist. Critical numbers are important because they give information about possible maximum or minimum points on the graph of a function.
To further understand this concept, let’s break down the two possible conditions for critical numbers:
1. When the derivative is equal to zero:
A critical number occurs when the derivative of a function is equal to zero at a specific point in its domain. This means that the slope of the function’s tangent line at that point is horizontal, or flat. At these points, the function may have a local maximum, a local minimum, or an inflection point. To determine which one it is, we often make use of the first and second derivative tests.
2. When the derivative does not exist:
A critical number can also occur when the derivative of a function is undefined at a particular point in its domain. This can happen if there is a vertical tangent line or a sharp corner at that specific point. At these points, it is crucial to examine the behavior of the function on both sides of the critical number to determine if there is a local maximum, a local minimum, or a point of discontinuity.
In summary, critical numbers play a significant role in identifying important features of a function, such as local extrema or points of inflection. By finding these critical numbers, we can gain valuable insights into the behavior of a function and its graph.
More Answers:
Understanding Decreasing Functions | The Relationship Between f'(x) and f(x)Understanding the Implications of f'(x) = 0 | Horizontal Tangent Lines and Local Extrema in Math Functions
Understanding Critical Points in Calculus | The Relationship Between Derivatives and Graphs