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

• What does the derivative $f'(x)$ indicate about the function $f(x)$?


  How $f(x)$ changes
• If $f''(x)$ is positive, then $f(x)$ is concave up
• What does the sign of $f'(x)$ indicate about $f(x)$?


  Increasing or decreasing
• When $f''(x)$ is zero, it indicates a potential inflection point.
• What does $f'(2) =$$- 3$ indicate about the tangent line at x = 2</latex>?


  It has a negative slope
• What information does the derivative f'(x)</latex> provide about the behavior of $f(x)$?


  Slope of tangent line
• Match the sign of $f''(x)$ with the concavity of $f(x)$:


  Positive ↔️ Concave up

  Negative ↔️ Concave down

  Zero ↔️ Potential inflection point
• Critical points occur where $f'(x)$ is zero or undefined.
• When $f''(x)$ is zero, it indicates a potential inflection point.
• If $f'(x)$ is positive, then $f(x)$ is increasing.


  True
• What does the second derivative $f''(x)$ measure about f(x)</latex>?


  Concavity
• What is the limit definition of the second derivative $f''(x)$?


  f''(x) = \lim_{h \to 0} \frac{f'(x + h) - f'(x)}{h}</latex>
• If $f'(x)$ is positive, then $f(x)$ is increasing.


  True
• If $f''(x)$ is negative, the graph of $f(x)$ is concave down.


  True
• Steps to analyze the concavity of a function using $f''(x)$:


  1️⃣ Find $f''(x)$

  2️⃣ Determine the intervals where $f''(x)$ is positive or negative

  3️⃣ Identify inflection points

  4️⃣ Conclude the concavity of $f(x)$ in each interval
• What does a positive value of $f'(x)$ indicate about $f(x)$?


  $f(x)$ is increasing
• What is the limit definition of the second derivative $f''(x)$?


  $f''(x) =$$\lim_{h \to 0} \frac{f'(x + h) - f'(x)}{h}$
• Match the sign of $f'(x)$ with the slope of the tangent line:


  Positive ↔️ Positive (upward) slope

  Negative ↔️ Negative (downward) slope

  Zero ↔️ Horizontal (flat) slope
• What are critical points defined as?

  $f'(x) =$$0$ or undefined
• Match the sign of $f''(x)$ with the concavity of $f(x)$:


  Positive ↔️ Concave up

  Negative ↔️ Concave down

  Zero ↔️ Potential inflection point
• If $f'(x)$ is positive, the tangent line to $f(x)$ has a positive slope
• If $f''(x)$ is positive, then $f(x)$ is concave up.


  True
• If $f'(x)$ is negative, the tangent line has a negative slope.
• Match the sign of $f'(x)$ with the behavior of $f(x)$:


  Positive ↔️ $f(x)$ is increasing

  Negative ↔️ $f(x)$ is decreasing

  Zero ↔️ $f(x)$ has a critical point
• What does $f''(x) > 0$ indicate about the graph of $f(x)$?


  Concave up
• If $f'(x) =$$0$, then $x$ is a critical point of $f(x)$.


  True
• The concavity of $f(x)$ is determined by the sign of $f''(x)$.


  True
• The concavity of a function determines whether it curves upwards or downwards
• If $f''(x)$ is negative, $f(x)$ is concave down.


  True
• Steps to determine inflection points using $f''(x)$


  1️⃣ Calculate the second derivative $f''(x)$

  2️⃣ Set $f''(x) =$$0$ or find where $f''(x)$ is undefined

  3️⃣ Check if $f''(x)$ changes sign around these points
• For $f(x) =$$x^{3} - 3x +$$1$, what is $f'(x)$?


  $3x^{2} - 3$
• What does the derivative $f'(x)$ of a function indicate?


  How $f(x)$ changes
• If $f'(x)$ is positive, then $f(x)$ is increasing
• If f'(x)</latex> is negative, the tangent line to $f(x)$ has a downward slope.


  True
• What does the second derivative $f''(x)$ measure?


  Rate of change of $f'(x)$
• The derivative $f'(x)$ represents the slope of the tangent line to the graph of $f(x)$ at $x$.


  True
• When f'(x)</latex> is zero, $f(x)$ has a critical point.
• When $f''(x)$ is zero and changes sign, it marks an inflection point.
• What does a negative value of $f'(x)$ indicate about $f(x)$?


  $f(x)$ is decreasing
• What does the second derivative $f''(x)$ measure?


  Rate of change of $f'(x)$