5.9 Connecting <latex>f'(x)</latex> and <latex>f''(x)</latex> with the Graph of <latex>f(x)</latex>

• The first derivative $f'(x)$ provides information about the behavior of $f(x)$, including the slope of the tangent line.
• If $f'(x) < 0$, then f(x)</latex> is decreasing.


  True
• Local maxima and minima occur at critical points where $f'(x) =$$0$ or is undefined.


  True
• The derivative $f'(x)$ represents the rate of change or slope of the function $f(x)$ at a given point.
• Match the condition on $f'(x)$ with the graph behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Upward sloping

  $f'(x) < 0$ ↔️ Downward sloping

  $f'(x) =$$0$ ↔️ Peak or valley
• A local maximum occurs when $f'(x)$ changes from positive to negative.


  True
• Match the condition on $f'(x)$ with the graph behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing

  $f'(x) =$$0$ ↔️ Local extrema
• If $f'(x) < 0$, then $f(x)$ is decreasing.


  True
• A local minimum occurs when $f'(x)$ changes from negative to positive.


  True
• What does a negative value of $f'(x)$ indicate about the graph of $f(x)$?


  Function is decreasing
• When $f'(x)$ changes sign, it indicates a local maximum or minimum
• Critical points occur where $f'(x) =$$0$ or is undefined.


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


  True
• When $f'(x) =$$0$, the graph of $f(x)$ has a horizontal tangent line.
• Steps to determine if a critical point is a local maximum or minimum using the First Derivative Test:

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

  2️⃣ Check the sign of $f'(x)$ to the left of the critical point

  3️⃣ Check the sign of $f'(x)$ to the right of the critical point

  4️⃣ If $f'(x)$ changes from positive to negative, it's a local maximum

  5️⃣ If $f'(x)$ changes from negative to positive, it's a local minimum
• Match the condition on f'(x)</latex> with the behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing

  $f'(x) =$$0$ ↔️ Horizontal tangent line

  $f'(x)$ changes sign ↔️ Local extrema
• Critical points occur where $f'(x) =$$0$ or is undefined and indicate local extrema.


  True
• The derivative $f'(x)$ indicates whether the function is increasing or decreasing.
• A local minimum occurs when $f'(x)$ changes from negative to positive.


  True
• If $f'(x) > 0$, the function $f(x)$ is increasing.
• A local maximum occurs when $f'(x)$ changes from positive to negative.


  True
• What does a positive value of $f'(x)$ indicate about the graph of $f(x)$?


  Function is increasing
• If f'(x) = 0</latex>, the graph of $f(x)$ has a horizontal tangent line.


  True
• What does the derivative f'(x)</latex> represent in terms of the function $f(x)$?


  Rate of change or slope
• What are critical points used to identify on the graph of $f(x)$?


  Local extrema
• A negative value of $f'(x)$ indicates that the graph is sloping downward
• Arrange the graph behaviors in order based on the sign of $f'(x)$:


  1️⃣ Increasing (f'(x) > 0)

  2️⃣ Decreasing (f'(x) < 0)

  3️⃣ Local Maximum (f'(x) = 0 or undefined)

  4️⃣ Local Minimum (f'(x) = 0 or undefined)
• If $f''(x) > 0$, the graph of $f(x)$ is concave up
• What is a point of inflection on the graph of $f(x)$?


  Change in concavity
• Arrange the conditions on $f'(x)$ in order based on the behavior of $f(x)$:


  1️⃣ $f'(x) > 0$ (Increasing)

  2️⃣ $f'(x) < 0$ (Decreasing)

  3️⃣ $f'(x) =$$0$ (Horizontal tangent line)
• What is the derivative $f'(x)$ for the function $f(x) =$$x^{2}$?


  $2x$
• If $f'(x) < 0$, what does it imply about $f(x)$?


  f(x) is decreasing
• For $f(x) =$$x^{2}$, $f'(x) =$$2x$


  True
• When $x < 0$, $f'(x) < 0$, so $f(x)$ is decreasing
• Match the condition on f'(x)</latex> with the behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing

  $f'(x) =$$0$ ↔️ Horizontal tangent line
• Critical points occur where $f'(x) =$$0$ or is undefined
• Match the sign of $f'(x)$ with the graph behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Upward sloping

  $f'(x) < 0$ ↔️ Downward sloping

  $f'(x) =$$0$ ↔️ Peak or valley
• If $f''(x) > 0$, the graph of $f(x)$ is concave up
• For $f(x) =$$x^{3}$, $f''(x) =$$6x$


  True
• Match the behavior of $f(x)$ with the sign of $f'(x)$:


  Increasing ↔️ Positive

  Decreasing ↔️ Negative

  Local extremum ↔️ Zero or Undefined