5.4 Using the First Derivative Test for Relative Extrema

• At what values of x does the function \( f(x) = x^3 - 3x + 2 \) have relative extrema?

  -1 and 1
• What are relative extrema of a function?

  Local maximum or minimum values
• A relative minimum occurs when the derivative changes from negative to positive.
• What type of relative extrema occurs at \( x = 2 \) for the function \( f(x) = x^2 - 4x + 3 \)?

  Relative Minimum
• A relative maximum occurs when the derivative changes from positive to negative.

  True
• What test is used to identify relative extrema in the function \( f(x) = x^3 - 3x + 2 \)?

  First Derivative Test
• Steps of the First Derivative Test

  1️⃣ Find Critical Points

  2️⃣ Create a Sign Chart

  3️⃣ Determine Relative Extrema
• What is the purpose of the First Derivative Test?

  Identify relative extrema
• Match the sign of \( f'(x) \) with the behavior of the function:

  Positive ↔️ Increasing

  Negative ↔️ Decreasing

  Zero ↔️ Critical Point
• What is the purpose of creating a sign chart in the First Derivative Test?

  Analyze function behavior
• Steps to apply the First Derivative Test to f(x) = x^2 - 4x + 3

  1️⃣ Find critical points: f'(x) = 2x - 4 = 0, so x = 2 is the critical point

  2️⃣ Create a sign chart

  3️⃣ Analyze the sign of f'(x) in intervals (-∞, 2) and (2, ∞)

  4️⃣ Determine relative extrema
• The First Derivative Test helps identify relative extrema by examining the behavior of the first derivative.

  True
• What are relative extrema?

  Local maximum or minimum
• The sign chart helps visualize the changes in the sign of the first derivative
• Relative extrema refer to local maximum and minimum values of a function
• The First Derivative Test finds relative extrema by analyzing the sign changes of the first derivative.


  True
• If f'(x) changes from positive to negative, there is a relative maximum.
• If f'(x) changes from negative to positive, there is a relative minimum at that critical point.


  True
• A relative maximum occurs when the derivative changes from positive to negative.

  True
• The First Derivative Test identifies relative extrema by analyzing the sign changes of the first derivative around a critical point.

  True
• Steps of the First Derivative Test

  1️⃣ Find Critical Points

  2️⃣ Create a Sign Chart

  3️⃣ Determine Relative Extrema
• Steps of the First Derivative Test

  1️⃣ Find Critical Points

  2️⃣ Create a Sign Chart

  3️⃣ Determine Relative Extrema
• The First Derivative Test involves finding critical points and creating a sign chart.
• If \( f'(x) \) changes from negative to positive, there is a relative minimum.
• Creating a sign chart involves placing critical points on a number line and testing the sign of \( f'(x) \) in each interval.
• If \( f'(x) \) changes from positive to negative at a critical point, the function has a relative maximum.


  True
• What are critical points in the context of the First Derivative Test?

  Values where f'(x) = 0
• If f'(x) is negative in an interval, the function is decreasing in that interval.


  True
• What type of relative extremum occurs if f'(x) changes from negative to positive?

  Relative minimum
• The function f(x) = x^2 - 4x + 3 has a relative minimum at x = 2
• The sign chart visualizes the changes in the sign of the first derivative, which corresponds to the increasing and decreasing behavior of the function
• If the first derivative, f'(x), is negative in an interval, the function is decreasing in that interval.


  True
• A relative minimum occurs when the first derivative changes from negative to positive.
• Steps to apply the First Derivative Test to find relative extrema:

  1️⃣ Find Critical Points

  2️⃣ Create a Sign Chart

  3️⃣ Determine Relative Extrema
• If f'(x) changes from positive to negative, there is a **relative maximum** at that critical point.
• Creating a sign chart involves testing the sign of f'(x) in each interval to analyze the function's behavior.
• In the interval (-∞, 2), the value of f'(x) is negative, indicating the function is decreasing.
• If f'(x) is positive in an interval, the function is increasing in that interval.
• At x = -1, the derivative f'(x) changes from positive to negative, indicating a relative maximum.
• To find critical points, set the first derivative equal to zero or determine where it is undefined.