Using the second derivative to determine concavity:

• What is concavity if the graph of a function bends upward?

  Concave upward
• Upward concavity occurs when the slope of the function is increasing.


  True
• Points of inflection occur where <img src="https://render.githubusercontent.com/render/svg?math=f''(x)"> is equal to zero or undefined
• What is the interpretation of the second derivative <img src="https://render.githubusercontent.com/render/svg?math=f''(x)">?

  Rate of change of slope
• A positive second derivative indicates upward concavity.


  True
• To find the second derivative, differentiate the first derivative twice
• If the second derivative of a function is negative, the function has downward concavity.


  True
• Match the condition with its concavity:

  $f''(x) > 0$ ↔️ Concave upward

  $f''(x) < 0$ ↔️ Concave downward

  $f''(x) =$$0$ ↔️ Potential point of inflection
• For the function $f(x) =$$x^{3} - 6x^{2} +$$5x - 10$, what is $f''(x)$?


  $6x - 12$
• For the function $f(x) =$$x^{3} - 6x^{2} +$$5x - 10$, when x < 2</latex>, $f''(x) < 0$ and the function is concave downward.


  True
• The first derivative test determines concavity based on the rate of change of f'(x).
• A function is concave upward if it curves like a cup.
• The second derivative indicates the rate of change of the slope of $f(x)$.
• If <img src="https://render.githubusercontent.com/render/svg?math=f''(x) > 0">, the function is concave upward
• If <img src="https://render.githubusercontent.com/render/svg?math=f''(x) > 0">, the function is concave upward
• If <img src="https://render.githubusercontent.com/render/svg?math=f''(x) < 0">, the function is concave downward
• If <img src="https://render.githubusercontent.com/render/svg?math=f'(x)"> is increasing, the function is concave upward
• If <img src="https://render.githubusercontent.com/render/svg?math=f''(x)"> changes sign around a point, it is a valid point of inflection.


  True
• $x =$$2$ is a point of inflection
• What condition must be met for a potential point of inflection to be valid?

  The sign of $f''(x)$ must change
• Steps to apply the second derivative test:

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

  2️⃣ Determine critical points by setting $f''(x) =$$0$ or finding undefined values

  3️⃣ Create intervals using critical points

  4️⃣ Choose a test value from each interval

  5️⃣ Evaluate $f''(x)$ for each test value

  6️⃣ Determine concavity based on the sign of $f''(x)$
• If $f''(x) > 0$, the function is concave upward.


  True
• Match the concavity with its definition:

  Upward ↔️ The function is curving upward

  Downward ↔️ The function is curving downward
• What does a negative second derivative imply about concavity?

  Concave downward
• What does the second derivative of a function indicate?

  Rate of change of slope
• If <img src="https://render.githubusercontent.com/render/svg?math=f''(x) < 0"/>, the function is concave downward.


  True
• What is the sign of the second derivative when a function is concave upward?

  Positive
• What does the second derivative <img src="https://render.githubusercontent.com/render/svg?math=f''(x)"> describe?

  Rate of change of slope
• When a function is curving upward, it is said to have upward concavity.
• If the second derivative of a function is positive, what is its concavity?

  Upward
• Steps to find the second derivative of a function

  1️⃣ Calculate the first derivative

  2️⃣ Differentiate the first derivative
• When $f''(x) < 0$, the function is concave downward.


  True
• For the function $f(x) =$$x^{3} - 6x^{2} +$$5x - 10$, what is $f'(x)$?


  $3x^{2} - 12x +$$5$
• What is the bending direction of a function when $f''(x) < 0$?


  Downward
• When $f''(x) < 0$, the function curves downward.


  True
• When $f'(x)$ is decreasing, the second derivative is negative.


  True
• If $f''(x) > 0$, the function is concave upward.


  True
• What does the second derivative of a function indicate?

  Rate of change of slope
• What does <img src="https://render.githubusercontent.com/render/svg?math=f''(x) = 0"> indicate about the function?

  Potential point of inflection
• What does <img src="https://render.githubusercontent.com/render/svg?math=f''(x) = 0"> indicate about the function?

  Potential point of inflection