5.4 Using the First Derivative Test for Relative Extrema

• A relative maximum at \( x = c \) occurs if \( f(c) \geq f(x) \) for all \( x \) in some interval around \( c \).


  True
• What is the primary purpose of the First Derivative Test?

  Identify relative extrema
• In the First Derivative Test, critical points are found where \( f'(x) = 0 \) or is undefined
• If \( f'(x) \) does not change sign at \( x = c \), there is neither a relative maximum nor a relative minimum.


  True
• If $f'(x)$ does not change sign at $x =$$c$, there is neither a relative maximum nor a relative minimum.


  True
• Steps of the First Derivative Test

  1️⃣ Find critical points

  2️⃣ Choose test values

  3️⃣ Evaluate $f'(x)$ at test values

  4️⃣ Determine relative extrema
• A sign change of $f'(x)$ from positive to negative indicates a relative maximum
• What are relative extrema characterized by?

  Sign changes in $f'(x)$
• The sign of $f'(x)$ is evaluated at test values in each interval to create a sign chart.


  True
• The second step in the First Derivative Test is to create a sign
• A relative minimum occurs at x = 3 because f'(x) changes from negative
• What are relative extrema also called?

  Local extrema
• A relative maximum occurs at x = c if f(c) ≥ f(x) for all x near c.


  True
• Match the sign change of f'(x) with the type of extrema:

  Positive to Negative ↔️ Relative Maximum

  Negative to Positive ↔️ Relative Minimum

  No Change ↔️ Neither
• Steps to determine relative minima using the First Derivative Test

  1️⃣ Find critical points where f'(x) = 0 or is undefined

  2️⃣ Create a sign chart and choose test values in each interval

  3️⃣ Evaluate f'(x) at each test value to find its sign

  4️⃣ Identify relative minima by looking for sign changes from negative to positive
• Critical points may represent potential relative extrema.


  True
• If f'(x) changes from positive to negative, there is a relative maximum
• In the sign chart, critical points are arranged on a number line.
• The derivative f'(x) of the example function is 3x^2 - 12x + 9.
• In the interval (1, 3), the sign of f'(x) is negative.


  True
• At x = 3, f'(x) changes from negative to positive, indicating a relative minimum.


  True
• The First Derivative Test uses the sign of \( f'(x) \) to determine if a critical point is a relative maximum, minimum, or neither
• Steps of the First Derivative Test

  1️⃣ Find critical points where \( f'(x) = 0 \) or is undefined

  2️⃣ Choose test values in intervals around each critical point

  3️⃣ Evaluate \( f'(x) \) at the test values

  4️⃣ Determine relative extrema based on sign changes
• What values of \( x \) are critical points of a function?

  \( f'(x) = 0 \) or undefined
• What is the first step of the First Derivative Test?

  Find critical points
• What is the First Derivative Test used to identify?

  Relative extrema
• If $f'(x)$ changes from positive to negative, there is a relative maximum.


  True
• A negative to positive sign change in f'(x)</latex> indicates a relative minimum.

  True
• To identify relative maxima, look for sign changes in $f'(x)$ from positive to negative
• What sign change indicates a relative minimum in the First Derivative Test?

  Negative to positive
• A relative minimum occurs where f'(x) changes from negative to positive.


  True
• Relative extrema are characterized by sign changes in f'(x).


  True
• What sign change in f'(x) indicates a relative maximum?

  Positive to Negative
• Steps to find relative minima using the First Derivative Test:

  1️⃣ Find critical points where f'(x) = 0 or is undefined

  2️⃣ Create a sign chart with critical points

  3️⃣ Evaluate f'(x) at test values in each interval

  4️⃣ Look for sign changes from negative to positive
• At what value of x does f'(x) change from positive to negative?

  x = 1
• Where does a relative minimum occur in the function f(x) = x^3 - 6x^2 + 9x + 1?

  x = 3
• In the function f(x) = x^3, why is x = 0 not a relative extremum?

  f'(x) does not change sign
• If f'(x) changes from negative to positive, there is a relative minimum.


  True
• If f'(x) does not change sign, there is neither a relative maximum nor a relative minimum.


  True
• What are the critical points of the example function?

  x = 1 and x = 3