5.5 Using the Second Derivative Test for Relative Extrema

Cards (82)

  • What is a relative extremum (or local extremum) of a function?
    Largest or smallest value
  • If \( f'(c) = 0 \) and \( f''(c) > 0 \), then \( f \) has a relative minimum at \( x = c \)

    True
  • If \( f'(c) = 0 \) and \( f''(c) = 0 \), the Second Derivative Test is inconclusive
  • What is the first derivative of \( f(x) = x^3 - 6x^2 + 9x + 1 \)?
    3x2−12x+3x^{2} - 12x +9 9
  • At the critical point \( x = 1 \) of \( f(x) = x^3 - 6x^2 + 9x + 1 \), there is a relative maximum because \( f''(1) < 0 \)

    True
  • The Second Derivative Test helps determine if a critical point is a relative maximum or relative minimum
  • What does it mean if \( f'(c) \) is undefined in the Second Derivative Test?
    Use the first derivative test
  • What is the value of \( f''(1) \) for \( f(x) = x^3 - 6x^2 + 9x + 1 \)?
    −6- 6
  • What conclusion can be drawn if \( f'(c) = 0 \) and \( f''(c) > 0 \)?
    Relative minimum at \( x = c \)
  • If \( f'(c) = 0 \) and \( f''(c) = 0 \), the second derivative test is inconclusive.
  • The function \( f(x) = x^3 - 6x^2 + 9x + 1 \) has a relative maximum at \( x = 1 \) because \( f''(1) = -6 < 0 \).
    True
  • What does the second derivative test evaluate to determine relative extrema?
    Sign of \( f''(c) \)
  • Relative extrema occur at critical points where the first derivative \( f'(x) \) is either zero or undefined
  • What type of extremum does a function have if \( f'(c) = 0 \) and \( f''(c) < 0 \)?
    Relative maximum
  • If \( f'(c) \) is undefined, \( x = c \) could be a relative extremum, and the first derivative test should be used.

    True
  • What is the second derivative of \( f(x) = x^3 - 6x^2 + 9x + 1 \)?
    6x−126x - 12
  • What type of extremum occurs at the critical point \( x = 3 \) of \( f(x) = x^3 - 6x^2 + 9x + 1 \)?
    Relative minimum
  • If \( f'(c) = 0 \) and \( f''(c) > 0 \), then \( f \) has a relative minimum at \( x = c \)

    True
  • Match the conditions with their conclusions in the Second Derivative Test:
    \( f'(c) = 0 \) and \( f''(c) > 0 \) ↔️ Relative minimum at \( x = c \)
    \( f'(c) = 0 \) and \( f''(c) < 0 \) ↔️ Relative maximum at \( x = c \)
    \( f'(c) = 0 \) and \( f''(c) = 0 \) ↔️ Test is inconclusive
  • The Second Derivative Test helps determine if a critical point is a relative maximum or relative minimum
  • The second derivative test helps determine whether a critical point c is a relative maximum or minimum.
  • If \( f'(c) = 0 \) and \( f''(c) < 0 \), the function has a relative maximum at \( x = c \).

    True
  • What should be used if \( f'(c) \) is undefined at \( x = c \)?
    First derivative test
  • The function \( f(x) = x^3 - 6x^2 + 9x + 1 \) has a relative minimum at \( x = 3 \) because \( f''(3) = 6 > 0.
  • Steps to apply the second derivative test to find relative extrema
    1️⃣ Find \( f'(x) \)
    2️⃣ Find critical points by setting \( f'(x) = 0 \)
    3️⃣ Find \( f''(x) \)
    4️⃣ Evaluate \( f''(c) \) at each critical point \( x = c \)
    5️⃣ Determine relative extrema based on the sign of \( f''(c) \)
  • Relative extrema occur at critical points where \( f'(x) \) is zero or undefined.
  • If \( f'(c) = 0 \) and \( f''(c) > 0 \), the function \( f(x) \) has a relative minimum at \( x = c \).
    True
  • What does \( f''(c) > 0 \) imply about the function \( f(x) \) at \( x = c \)?
    Relative minimum at \( x = c \)
  • What does the condition \( f'(c) = 0 \) and \( f''(c) > 0 \) indicate about \( f \) at \( x = c \)?
    Relative minimum
  • When \( f''(c) > 0 \), the Second Derivative Test indicates a relative minimum at \( x = c \).

    True
  • The relative minimum of \( f(x) = x^2 - 4x + 6 \) occurs at the point \( (2, 2) \).
  • What condition implies that \( f \) has a relative maximum at \( x = c \)?
    \( f'(c) = 0 \) and \( f''(c) < 0 \)
  • Match the condition with the conclusion for the Second Derivative Test:
    \( f'(c) = 0 \) and \( f''(c) > 0 \) ↔️ Relative minimum at \( x = c \)
    \( f'(c) = 0 \) and \( f''(c) < 0 \) ↔️ Relative maximum at \( x = c \)
    \( f'(c) = 0 \) and \( f''(c) = 0 \) ↔️ Test is inconclusive
  • For \( f(x) = x^3 - 6x^2 + 9x + 1 \), the relative maximum occurs at \( x = 1) \).
  • Steps to apply the Second Derivative Test when \( f''(c) > 0 \):
    1️⃣ Find \( f'(x) \) and \( f''(x) \)
    2️⃣ Determine critical points by solving \( f'(x) = 0 \)
    3️⃣ Evaluate \( f''(c) \) at each critical point
    4️⃣ If \( f''(c) > 0 \), conclude relative minimum at \( x = c \)
  • When \( f''(c) > 0 \), the Second Derivative Test indicates a relative minimum at \( x = c \).

    True
  • What is the value of \( f''(3) \) for \( f(x) = -x^2 + 6x - 5 \)?
    -2
  • The condition \( f'(c) = 0 \) and \( f''(c) < 0 \) indicates a relative maximum at \( x = c \).
    True
  • What is the value of \( f''(3) \) for \( f(x) = -x^2 + 6x - 5 \)?
    -2
  • If \( f'(c) = 0 \) and \( f''(c) < 0 \), the function has a relative maximum at \( x = c \).
    True