5.3 Determining Intervals on Which a Function is Increasing or Decreasing

• What happens to a function if its derivative is negative on an interval?

  It is decreasing
• What are critical points of a function?

  Values where f'(x) = 0 or undefined
• What is the first step in determining the sign of the derivative in each interval?

  Find the critical points
• A positive derivative indicates that the function is increasing.


  True
• What does a derivative of zero indicate about the function at that point?

  May be increasing, decreasing, or constant
• For $x < 0$, the derivative of $f(x) =$$x^{3} - 6x^{2} +$$5$ is negative.


  False
• For $x > 4$, the derivative of $f(x) =$$x^{3} - 6x^{2} +$$5$ is negative
• If the derivative is positive, the function is increasing.


  True
• Arrange the intervals of increasing and decreasing behavior of f(x) = x^{3} - 6x^{2} + 5</latex> in order:

  1️⃣ Decreasing for $x < 0$

  2️⃣ Increasing for $0 < x < 4$

  3️⃣ Decreasing for $x > 4$
• The first derivative test states that if $f'(x) > 0$, then $f(x)$ is increasing
• What is the derivative of \( f(x) = x^3 - 6x^2 + 5 \)?

  $f'(x) =$$3x^{2} - 12x$
• What is the first step in finding critical points of a function?

  Compute the derivative
• What do you do after creating intervals using critical points in the first derivative test?

  Choose test values
• What does a zero value of the derivative indicate in the first derivative test?

  Potential local max/min
• The function \( f(x) = x^3 - 6x^2 + 5 \) is decreasing on the interval $(0, 4)$.


  True
• If the derivative of a function is positive on an interval, the function is increasing on that interval

  True
• Steps of the first derivative test to determine increasing and decreasing intervals

  1️⃣ Find the critical points of the function

  2️⃣ Determine the sign of the derivative in each interval

  3️⃣ Use the first derivative test to identify intervals

  4️⃣ Write the intervals where the function is increasing and decreasing
• To determine the sign of the derivative in each interval, the first step is to find the critical points
• When the derivative is negative, the function is decreasing
• What is the derivative of $f(x) =$$x^{3} - 6x^{2} +$$5$?


  $f'(x) =$$3x^{2} - 12x$
• In which interval is the derivative of $f(x) =$$x^{3} - 6x^{2} +$$5$ positive?


  $0 < x < 4$
• Critical points are found where the derivative is equal to zero
• For $x < 0$, the derivative of $f(x) =$$x^{3} - 6x^{2} +$$5$ is negative, so the function is decreasing
• Match the derivative sign with the function behavior:

  Positive ↔️ Increasing

  Negative ↔️ Decreasing

  Zero ↔️ May be constant
• Steps to find the critical points of a function:

  1️⃣ Compute the derivative

  2️⃣ Set the derivative to zero

  3️⃣ Solve for x

  4️⃣ Identify undefined points
• Critical points of a function occur where the derivative is zero or undefined.


  True
• The first derivative test helps determine whether a function is increasing or decreasing in each interval
• Match the derivative sign with the function behavior:

  Positive derivative ↔️ Increasing function

  Negative derivative ↔️ Decreasing function

  Zero derivative ↔️ Potential local max/min
• Steps to use the first derivative test to find increasing and decreasing intervals

  1️⃣ Find the derivative \( f'(x) \)

  2️⃣ Find the critical points by setting \( f'(x) = 0 \)

  3️⃣ Create intervals using the critical points

  4️⃣ Test a value within each interval to determine the sign of \( f'(x) \)
• The sign of the derivative of a function determines whether the function is increasing or decreasing
• If the derivative of a function is zero at a point, the function may be increasing, decreasing, or constant
• The critical points of $f(x) =$$x^{3} - 6x^{2} +$$5$ are $x =$$0$ and $x =$$4$


  True
• The first derivative test is used to determine the sign of the derivative in each interval
• When the derivative is positive, the function is increasing
• Steps to identify intervals and determine the sign of the derivative

  1️⃣ Compute the derivative

  2️⃣ Set the derivative equal to zero

  3️⃣ Solve for critical points

  4️⃣ Divide the x-axis into intervals

  5️⃣ Determine the sign of the derivative in each interval
• The function $f(x) =$$x^{3} - 6x^{2} +$$5$ is decreasing for $x < 0$.


  True
• The critical points of f(x) = x^{3} - 6x^{2} + 5</latex> are $x =$$0$ and $x =$$4$.


  True
• If $f'(x) > 0$, then $f(x)$ is increasing.


  True
• A critical point occurs when the derivative is either zero or undefined.

  True
• The critical points of \( f(x) = x^3 - 6x^2 + 5 \) are found by solving \( 3x^2 - 12x = 0 \), which gives \( x = 0 \) and \( x = 4