5.5 Using the Second Derivative Test for Relative Extrema

• What are relative extrema in calculus?

  Local maximum and minimum
• A relative maximum occurs when $f(c) \geq f(x)$ for all $x$ near $c$.
• Relative extrema occur at critical points where $f'(x) =$$0$ or $f'(x)$ is undefined
• Match the characteristic with the type of relative extrema:

  $f''(c) < 0$ ↔️ Relative Maximum

  $f''(c) > 0$ ↔️ Relative Minimum
• What does the second derivative represent in calculus?

  Rate of change of slope
• If $f''(x) > 0$, the function is concave up.
• A point of inflection occurs when $f''(x) =$$0$ and concavity changes.
• Match the second derivative sign with the concavity:

  f''(x) > 0</latex> ↔️ Concave Up

  $f''(x) < 0$ ↔️ Concave Down

  $f''(x) =$$0$ ↔️ Point of Inflection
• What does $f''(x) =$$0$ indicate about the concavity of a function?


  Potential change in concavity
• The first derivative $f'(x)$ determines the concavity
• If $f''(x) > 0$, the function is concave up.
• If $f''(x) < 0$, the function is concave down.
• The second derivative is denoted as $f''(x)$ and is the derivative of the first derivative $f'(x)$ to determine concavity
• If $f''(x) =$$0$, it indicates a possible point of inflection.
• Steps to find relative extrema using the second derivative test

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

  2️⃣ Determine critical points by setting $f'(x) =$$0$

  3️⃣ Find the second derivative $f''(x)$

  4️⃣ Evaluate $f''(c)$ for each critical point $c$
• If $f''(c) > 0$, then $f(c)$ is a relative minimum
• If f''(c) = 0</latex>, the second derivative test is inconclusive.
• If $f''(c) > 0$, what is $f(c)$?


  Relative minimum
• If f''(c) <0</latex>, then $f(c)$ is a relative maximum.
• What happens if $f''(c) =$$0$?


  Test is inconclusive
• Steps to find relative extrema using the second derivative test

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

  2️⃣ Determine the critical points by setting $f'(x) =$$0$

  3️⃣ Find the second derivative $f''(x)$

  4️⃣ Evaluate $f''(c)$ for each critical point $c$
• If $f''(c) > 0$, then $f(c)$ is a relative minimum.
• For $f(x) =$$x^{3} - 6x^{2} +$$5$, what is the value of $f''(0)$?


  -12
• For $f(x) =$$x^{3} - 6x^{2} +$$5$, f''(4) =12</latex> indicates a relative minimum
• What are relative extrema in calculus?

  Local high and low points
• Define a relative maximum.

  $f(c) \geq f(x)$ for all $x$ near $c$
• What does the second derivative represent?

  Rate of change of slope
• If $f''(x) > 0$, the function is concave up.
• If f''(x) < 0</latex>, the function is concave down
• What does $f''(x) =$$0$ indicate?


  Potential point of inflection
• For f(x) = x^{3}</latex>, $f'(x) =$$3x^{2}$, and $f''(x) =$$6x$, is the function concave up for $x > 0$?


  Yes
• What is one condition for using the second derivative test?

  Function twice differentiable
• A critical point $c$ for the second derivative test requires f'(c) = 0
• If $f''(c) < 0$, what is the result of the second derivative test?


  Relative maximum
• If $f''(c) > 0$, what is the result of the second derivative test?


  Relative minimum
• Steps to apply the second derivative test to find relative extrema:

  1️⃣ Find the first derivative $f'(x)$ and determine the critical points by setting $f'(x) =$$0$

  2️⃣ Find the second derivative $f''(x)$

  3️⃣ Evaluate $f''(c)$ for each critical point $c$
• Match the sign of $f''(c)$ with its result in the second derivative test:


  Positive ↔️ Relative minimum at $f(c)$

  Negative ↔️ Relative maximum at $f(c)$

  Zero ↔️ Test is inconclusive
• A positive value of $f''(c)$ indicates a relative minimum at $f(c)$
• If $f''(c) =$$0$, the second derivative test is inconclusive
• Interpreting the results of the second derivative test involves analyzing the sign of $f''(c)$ at critical points to determine if the point is a relative maximum, relative minimum, or inconclusive