Using the second derivative to determine concavity:

• What does concavity refer to in the context of a graph?

  The shape of a curve
• What is the shape of a curve when it is concave down?

  An upside-down bowl
• What is the first derivative of $f(x) =$$x^{3} - 3x^{2} +$$5$?


  $3x^{2} - 6x$
• What rules are used to find the first derivative of a function?

  Differentiation rules
• Critical points occur where the second derivative is equal to zero or undefined.


  True
• The second derivative can be undefined at points where the concavity may change.


  True
• Concavity refers to the shape of a curve on a graph.

  True
• Match the concavity with its shape:

  Concave Up ↔️ U

  Concave Down ↔️ ∩
• Steps to find the second derivative of a function:

  1️⃣ Find the first derivative

  2️⃣ Find the second derivative by differentiating the first

  3️⃣ Simplify the second derivative
• Critical points of the second derivative may occur where the second derivative is undefined.


  True
• If $f''(x) > 0$ in an interval, the function is concave up.


  True
• Steps to determine concavity in intervals based on the second derivative

  1️⃣ Identify intervals based on critical points

  2️⃣ Choose test values within each interval

  3️⃣ Evaluate $f''(x)$ at each test point

  4️⃣ Interpret the sign of $f''(x)$ to determine concavity
• Match the concavity with the sign of $f''(x)$:


  Concave Up ↔️ +

  Concave Down ↔️ -
• If $f''(x) > 0$, the function is concave up.


  True
• A curve that looks like an upside-down bowl is called concave down.
• What is the first step to find the second derivative of a function?

  Find the first derivative
• To find the critical points of the second derivative, you set $f''(x)$ equal to zero.
• If f''(0) = - 6</latex>, the function is concave down at $x =$$0$.


  True
• A positive sign of the second derivative indicates the function is concave up
• For $x =$$0$, $f''(0) =$$- 6$, indicating the function is concave down
• To determine concavity in different intervals, we look at the sign of the second derivative
• How are critical points $a$ and $b$ found for concavity analysis?


  $f''(x) =$$0$
• Steps to identify points of inflection

  1️⃣ Find $f''(x)$

  2️⃣ Set $f''(x) =$$0$ and solve for $x$

  3️⃣ Check where $f''(x)$ is undefined

  4️⃣ Verify that the sign of $f''(x)$ changes across each critical point
• A concave up curve has a shape that resembles the letter U
• Steps to find the second derivative of a function

  1️⃣ Find the first derivative

  2️⃣ Find the derivative of the first derivative

  3️⃣ Simplify the expression
• The function $f(x) =$$x^{2}$ is an example of a curve that is always concave up
• What does setting the second derivative equal to zero help us find?

  Potential inflection points
• What are critical points of the second derivative called?

  Points of concavity change
• What do critical points and points of undefined second derivative help determine?

  Concavity intervals
• What happens to the derivative when a curve is concave down?

  It is decreasing
• What is an example of a function that is concave up?

  $f(x) =$$x^{2}$
• The second derivative of $f(x) =$$x^{3} - 3x^{2} +$$5$ is 6x - 6
• If $f''(x) =$$6x - 6$, what is the sign of $f''(0)$?


  Negative
• To determine the concavity of a function, we look at the sign of the second derivative.
• If $f''(x) < 0$, the function is concave down.


  True
• To determine the concavity of a function, we look at the sign of the second derivative.
• Match the concavity with the sign of $f''(x)$:


  Concave Up ↔️ +

  Concave Down ↔️ -
• A curve that looks like a bowl opening upwards is called concave up.
• The second derivative of $f(x) =$$x^{3} - 3x^{2} +$$5$ is $f''(x) =$$6x - 6$.


  True
• What does the sign of the second derivative indicate in an interval?

  The concavity