Applying the Second Derivative Test:

• What does a negative second derivative at a critical point indicate?

  Relative maximum
• Steps to find the critical points of a function f(x)

  1️⃣ Compute the first derivative f'(x)

  2️⃣ Set f'(x) = 0 and solve for x

  3️⃣ Check where f'(x) is undefined

  4️⃣ List all x values
• For f(x) = x² - 4x + 3, what is f'(x)?

  2x - 4
• If f''(x) < 0 at a critical point, the function has a relative maximum
• What type of extremum does the critical point x = 2 have for f(x) = x² - 4x + 3?

  Relative minimum
• For f(x) = x² - 4x + 3, the first derivative f'(x) is 2x
• The second derivative is calculated by differentiating the first derivative.


  True
• The second derivative measures the rate of change of the slope
• Given f(x) = x^3 - 3x^2 + 5x - 2, what is f'(x)?

  3x^2 - 6x + 5
• The Second Derivative Test can determine whether a critical point is a relative maximum, minimum, or inconclusive.

  True
• If f''(2) = 6 > 0, then x = 2 is a relative minimum
• What is the first derivative of the function f(x) = x^2 - 4x + 3?

  f'(x) = 2x - 4
• What is the only critical point of the function f(x) = x^2 - 4x + 3?

  x = 2
• Steps to calculate the second derivative

  1️⃣ Find the first derivative f'(x)

  2️⃣ Differentiate f'(x) to get f''(x)
• What is the value of f''(2) for the function f(x) = x^3 - 3x^2 + 5x - 2?

  f''(2) = 6
• If f''(x) > 0 at a critical point, it indicates a relative maximum.

  False
• To find the critical points of a function, we need to find the values of x where the first derivative f'(x) is zero or undefined
• Critical points occur only where f'(x) = 0.

  False
• A positive second derivative at a critical point indicates a relative minimum.

  True
• Steps to find the critical points of a function f(x)

  1️⃣ Find f'(x)

  2️⃣ Set f'(x) = 0 and solve for x

  3️⃣ Check where f'(x) is undefined

  4️⃣ List all critical points
• Match the condition with the relative extremum:

  f''(x) < 0 ↔️ Relative maximum

  f''(x) > 0 ↔️ Relative minimum

  f''(x) = 0 ↔️ Test inconclusive
• Critical points are found by setting the first derivative equal to zero
• The second derivative of f(x) = x^3 - 3x^2 + 5x - 2 is f''(x) = 6x - 6
• Match the condition with the relative extremum:

  f''(x) < 0 ↔️ Relative maximum

  f''(x) > 0 ↔️ Relative minimum

  f''(x) = 0 ↔️ Inconclusive
• Steps to find the critical points of a function

  1️⃣ Find the first derivative f'(x)

  2️⃣ Set f'(x) = 0 and solve for x

  3️⃣ Identify undefined points

  4️⃣ List critical points
• To find the critical points, the first derivative must be set equal to zero
• The Second Derivative Test is used to determine whether a critical point is a relative maximum
• What does it mean if f''(x) = 0 at a critical point?

  Test is inconclusive
• What is the purpose of the Second Derivative Test?

  Determine relative extrema
• For f(x) = x² - 4x + 3, what is f''(x)?

  2
• What is the critical value for the function f(x) = x² - 4x + 3?

  x = 2
• What does the second derivative measure?

  Concavity of the function
• You must check where the first derivative is undefined to find all critical points.


  True
• Steps to calculate the second derivative

  1️⃣ Find the first derivative f'(x)

  2️⃣ Differentiate f'(x) to get f''(x)
• What is the first derivative of f(x) = x^3 - 3x^2 + 5x - 2?

  3x^2 - 6x + 5
• If f'(x) is defined for all x, then the only critical point of f(x) = x^2 - 4x + 3 is x = 2
• The first derivative f'(x) = 2x - 4 is defined for all x.

  True
• What is the first derivative of f(x) = x^3 - 3x^2 + 5x - 2?

  f'(x) = 3x^2 - 6x + 5
• Since f''(2) = 6 > 0, the function has a relative minimum at x = 2.


  True
• Steps to determine whether a critical point is a relative maximum or minimum using the Second Derivative Test

  1️⃣ Calculate f''(x)

  2️⃣ Evaluate f''(x) at each critical point

  3️⃣ Interpret the sign of f''(x)