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) *$$*$ indicates the slope of the tangent to $f(x)$ at any given point
• If $*$$*$$f'(x) > 0 *$$*$, then $f(x)$ is increasing at $x$.
• If * * f'(x) <0 * *</latex>, then $f(x)$ is decreasing
• If $*$$*$$f'(x) =$$0 *$$*$, then $f(x)$ has a critical point at $x$.
• Match the condition with the behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing

  $f'(x) =$$0$ ↔️ Local minimum, maximum, or inflection point
• The first derivative * * f'(x) * *</latex> indicates the slope of the tangent to $f(x)$ at any given point.
• If $*$$*$$f'(x) > 0 *$$*$, then $f(x)$ is increasing at $x$.
• If $*$$*$$f'(x) < 0 *$$*$, then $f(x)$ is decreasing
• If $*$$*$$f'(x) =$$0 *$$*$, then $f(x)$ has a local minimum, maximum, or inflection point.
• Steps to analyze the behavior of $f(x)$ using $f'(x)$:


  1️⃣ Find $*$$*$$f'(x) *$$*$

  2️⃣ Set $*$$*$$f'(x) =$$0 *$$*$ to find critical points

  3️⃣ Determine intervals where $*$$*$$f'(x) > 0 *$$*$ or $*$$*$$f'(x) < 0 *$$*$

  4️⃣ Identify increasing or decreasing behavior of $f(x)$
• If $*$$*$$f'(x) > 0 *$$*$, then $f(x)$ is increasing at $x$.
• If $*$$*$$f'(x) > 0 *$$*$, then $f(x)$ is increasing
• If $*$$*$$f'(x) < 0 *$$*$, then f(x)</latex> is decreasing at $x$.
• If $*$$*$$f'(x) =$$0 *$$*$, then $f(x)$ has a local minimum, maximum, or inflection point.
• Match the condition with the behavior of $f(x)$:


  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing

  $f'(x) =$$0$ ↔️ Local minimum, maximum, or inflection point
• The second derivative * * f''(x) * *</latex> describes the concavity of a function $f(x)$.
• If $*$$*$$f''(x) > 0 *$$*$, then $f(x)$ is concave up.
• If $*$$*$$f''(x) < 0 *$$*$, then $f(x)$ is concave down
• If $*$$*$$f''(x) =$$0 *$$*$, $f(x)$ may have an inflection point.
• Match the condition with the concavity of $f(x)$:


  f''(x) > 0</latex> ↔️ Concave up

  $f''(x) < 0$ ↔️ Concave down

  $f''(x) =$$0$ ↔️ Inflection point possible
• If $*$$*$$f''(x) > 0 *$$*$, then $f(x)$ is concave up.
• If $*$$*$$f''(x) < 0 *$$*$, then $f(x)$ is concave down
• If $*$$*$$f''(x) =$$0 *$$*$, $f(x)$ may have an inflection point.
• What type of point may exist when $f''(x) =$$0$?


  Inflection point
• If $f''(x) > 0$, then $f(x)$ is concave up at $x$.
• If $f''(x) < 0$, then f(x)</latex> is concave down at $x$.
• For $f(x) =$$x^{3}$, when is $f''(x) > 0$?


  $x > 0$
• If $f''(x) < 0$, what can you conclude about the concavity of $f(x)$?


  Concave down
• If $f''(x) =$$0$, then $f(x)$ must have an inflection point at $x$.


  False
• A positive second derivative $f''(x) > 0$ indicates that f(x)</latex> is concave
• For $f(x) =$$x^{3}$, what is $f''(x)$?


  $6x$
• The second derivative $f''(x)$ indicates the concavity of $f(x)$.
• If $f''(x) > 0$, then $f(x)$ is concave up at $x$.
• Match the condition of $f''(x)$ with the behavior of $f(x)$:


  $f''(x) > 0$ ↔️ Concave up

  $f''(x) < 0$ ↔️ Concave down

  $f''(x) =$$0$ ↔️ Possible inflection point
• For $f(x) =$$x^{3}$, at what values of $x$ is $f(x)$ concave up and concave down?


  Up: $x > 0$, Down: $x < 0$
• The first derivative $f'(x)$ represents the slope of the tangent to $f(x)$.
• If $f'(x) > 0$, then $f(x)$ is increasing at $x$.
• For $f(x) =$$x^{2} - 4x +$$3$, what is $f'(x)$?


  $2x - 4$
• At $x =$$2$, $f'(x) =$$0$ for $f(x) =$$x^{2} - 4x +$$3$, indicating a local minimum at $(2, - 1)$.
• Critical points of $f(x)$ occur only where $f'(x) =$$0$.


  False