Calculus I

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First derivative test

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Calculus I

Definition

The first derivative test is a method used to determine the local maxima and minima of a function by analyzing the sign changes of its first derivative. It helps in understanding the increasing and decreasing behavior of the function.

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5 Must Know Facts For Your Next Test

  1. A critical point is where the first derivative, $f'(x)$, is zero or undefined.
  2. If $f'(x)$ changes from positive to negative at a critical point, then $f(x)$ has a local maximum at that point.
  3. If $f'(x)$ changes from negative to positive at a critical point, then $f(x)$ has a local minimum at that point.
  4. If there is no sign change in $f'(x)$ around a critical point, it implies that there is neither a maximum nor minimum at that point.
  5. The first derivative test can also help determine intervals where the function is increasing or decreasing.

Review Questions

  • What does it mean if $f'(x)$ changes from positive to negative at a critical point?
  • How do you find the critical points of a function?
  • What conclusion can be drawn if there is no sign change in $f'(x)$ around a critical point?
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