5.4 Using the First Derivative Test for Relative Extrema

• Critical points are identified by finding where the first derivative, f'(x), is equal to 0 or undefined.
• Steps to identify critical points

  1️⃣ Find points where f'(x) = 0

  2️⃣ Find points where f'(x) is undefined
• If f'(x) > 0, the function is increasing.
• Match the sign change in f'(x) with the relative extrema type:

  + to - ↔️ Local Maximum

  to + ↔️ Local Minimum

  + to + or - to - ↔️ No Relative Extrema
• What happens to the function behavior at a local maximum based on the sign change of its first derivative?

  Increases then decreases
• When the sign of the first derivative does not change at a critical point, the function has no relative extrema
• Critical points are always relative extrema.

  False
• What does the sign of f'(x) indicate about the function?

  Whether it is increasing
• If f'(x) = 0, the function is neither increasing nor decreasing.

  True
• If f'(x) changes from + to - at a critical point, it is a relative maximum.
• What happens at a critical point if f'(x) does not change sign?

  No relative extrema
• If the first derivative changes from - to + at a critical point, it indicates a local minimum.


  True
• What does the First Derivative Test state about the sign change of f'(x) at a relative maximum?

  Changes from + to -
• If f'(x) does not change sign at a critical point, it is neither a relative maximum nor a relative minimum.


  True
• Summarize the function behavior at a critical point where f'(x) changes from + to -.

  Increases then decreases
• Steps to identify critical points

  1️⃣ Find points where f'(x) = 0

  2️⃣ Find points where f'(x) is undefined
• A local maximum occurs at a critical point where f'(x) is equal to 0
• After finding critical points, the sign of f'(x) is determined in intervals around them to understand whether the function is increasing
• If f'(x) changes from + to - at a critical point, the function behavior is decreasing.

  False
• For the function f(x) = x² - 4x + 3, f'(x) changes from negative to positive at x = 2, indicating a relative minimum
• If f'(x) > 0 in an interval, the function is increasing in that interval.

  True
• Steps to determine the sign of f'(x)

  1️⃣ Choose a test value in each interval

  2️⃣ Evaluate f'(x) at each test value

  3️⃣ Analyze the sign of f'(x)
• For x < 2, f'(x) = 2x - 4 is negative, indicating the function is decreasing
• What test is used to classify critical points based on the sign of f'(x)?

  First Derivative Test
• If f'(x) changes from - to + at a critical point, it is classified as a relative minimum.

  True
• Use critical points to divide the domain into intervals
• What are the critical points of f(x) = x³ - 3x² + 2?

  x = 0 and x = 2
• What type of critical point occurs at x = 2 for f(x) = x³ - 3x² + 2?

  Relative Minimum
• What type of critical point occurs when f'(x) changes from - to +?

  Relative minimum
• For x₁ = 2, if f'(x) changes from + to -, then x₁ is a relative maximum
• What are critical points defined as in calculus?

  f'(x) = 0 or undefined
• Critical points indicate the locations of relative extrema.

  True
• What condition must be met for a point of inflection to exist?

  f'(x) is undefined
• Steps to determine the sign of f'(x) around critical points

  1️⃣ Choose a test value in each interval

  2️⃣ Evaluate f'(x) at each test value

  3️⃣ Analyze the sign of f'(x)
• Consider f(x) = x² - 4x + 3. What is f'(x)?

  2x - 4
• To understand whether a function is increasing or decreasing around critical points, you need to determine the sign of its first derivative
• What happens to the function if f'(x) < 0 in an interval?

  It is decreasing
• Match the sign change in f'(x) with the type of relative extrema and function behavior

  + to - ↔️ Relative Maximum, Increases then decreases

  to + ↔️ Relative Minimum, Decreases then increases

  No change ↔️ Neither Relative Max nor Min, Function is monotonic
• If f'(x) changes from negative to positive at x = 2, it is a relative minimum.

  True
• A sign change in f'(x) from positive to negative at a critical point indicates a relative maximum