5.3 Determining Intervals on Which a Function is Increasing or Decreasing

• If $f'(x) > 0$, then the function is increasing
• Match the condition with the function behavior:

  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing

  $f'(x) =$$0$ ↔️ May be increasing, decreasing, or constant
• The power rule for differentiation is d/dx (x^n) = nx^{n-1}</latex>, where `n` is a constant
• What is the derivative of $f(x) =$$x^{2}$?


  $f'(x) =$$2x$
• Critical points occur only where the derivative is equal to zero.

  False
• Critical points occur where the derivative is equal to zero
• Why are critical points important in analyzing a function's behavior?

  May change function behavior
• The derivative of a function represents its rate
• What does f'(x) < 0 indicate about the function's behavior?

  Function is decreasing
• Match the derivative condition with the corresponding function behavior.

  f'(x) > 0 ↔️ Increasing

  f'(x) < 0 ↔️ Decreasing

  f'(x) = 0 ↔️ Constant
• The power rule states that d/dx (x^n) = nx^(n-1)
• What is the derivative of 2x^2 using the constant multiple rule?

  4x
• Steps to find the derivative of a function.

  1️⃣ Identify the function

  2️⃣ Apply differentiation rules

  3️⃣ Simplify the expression
• Critical points occur where the function's behavior may change, such as from increasing to decreasing
• What are the critical points of f(x) = x^3 + 2x^2 - 5x + 1?

  x = -2.33, x = 0.67
• Critical points occur when the derivative is equal to zero
• What do critical points indicate about a function's behavior?

  Change in behavior
• The critical points of the function f(x) = x^3 + 2x^2 - 5x + 1</latex> are $x =$$- 2$ and $x =$$1$.


  True
• If f'(x) > 0</latex> within an interval, the function is increasing
• What should you do after creating intervals with critical points to determine function behavior?

  Test derivative values
• To test values, we compute the derivative `f'(x)` at the test value
• If `f'(x) = 0`, the function has a critical point.


  True
• If `f'(x) > 0`, then the function is increasing
• To find the derivative of a function, apply the appropriate differentiation rules
• The derivative of `f(x) = x^3 + 2x^2 - 5x + 1` is `f'(x) = 3x^2 + 4x - 5
• At a local maximum, `f'(x) = 0` and `f''(x) < 0`.


  True
• If `f'(x) > 0` in an interval, the function is increasing in that interval.


  True
• If f'(x) = 0 at a point, it indicates a critical point
• Steps to analyze intervals created by critical points

  1️⃣ Create Intervals: Divide the number line using critical points.

  2️⃣ Test Derivative Values: Choose a test value within each interval and evaluate f'(x).

  3️⃣ Determine Function Behavior: If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing; if f'(x) = 0, there is a critical point.
• If f'(x) = 0 at a point, there is a critical point.


  True
• What are the critical points of f(x) = x^3 - 6x^2 + 9x - 4?

  x = 1 and x = 3
• What is the sign of f'(x) in the interval (1, 3) for f(x) = x^3 - 6x^2 + 9x - 4?

  Negative
• What does it mean if f'(x) = 0 in an interval?

  There is a critical point
• The critical points are used to create intervals
• The function f(x) = x^3 - 6x^2 + 9x - 4 is increasing in the interval (-∞, 1).


  True
• Steps to determine if a function is increasing or decreasing

  1️⃣ Create Intervals: List all critical points to divide the number line into intervals.

  2️⃣ Select Test Values: Choose one value within each interval.

  3️⃣ Evaluate Derivative: Compute the derivative f'(x) at the test value.

  4️⃣ Determine Function Behavior: If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing; if f'(x) = 0, there is a critical point.
• What does it mean if f'(x) > 0 in an interval when testing derivative values?

  The function is increasing
• If f'(x) = 0, the function has a critical point
• The function f(x) = x^3 - 6x^2 + 9x - 4 is decreasing in the interval (1, 3).


  True
• Steps to determine if a function is increasing or decreasing

  1️⃣ Create Intervals: List all critical points to divide the number line into intervals.

  2️⃣ Select Test Values: Choose one value within each interval.

  3️⃣ Evaluate Derivative: Compute the derivative f'(x) at the test value.

  4️⃣ Determine Function Behavior: If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing; if f'(x) = 0, there is a critical point.