Applying the Second Derivative Test:

• What does the Second Derivative Test determine about a function at a critical point?

  Relative extrema
• If $f''(c) > 0$, then $f(x)$ has a local minimum at $x =$$c$.


  True
• The derivative of $x^{n}$ is nx^{n - 1}.
• What is the Second Derivative Test used for?

  Determining relative extrema
• If $f''(c) =$$0$, the Second Derivative Test is inconclusive
• What does the first derivative of a function represent?

  Slope of the tangent line
• What is the first derivative of $f(x) =$$2x^{3} - 5x^{2} +$$7x - 3$?


  $6x^{2} - 10x +$$7$
• What does the second derivative of a function represent?

  Rate of change of the slope
• Steps to find critical points of a function

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

  2️⃣ Set $f'(x) =$$0$

  3️⃣ Solve for $x$
• To determine the nature of extrema, you evaluate the second derivative $f''(x)$ at each critical point
• What should you do if $f''(c) =$$0$?


  Use the First Derivative Test
• What does the first derivative $f'(x)$ of a function represent?


  Slope of the tangent line
• Steps to apply the Second Derivative Test

  1️⃣ Find critical points by solving $f'(x) =$$0$

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

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

  4️⃣ Determine relative extrema based on the sign of $f''(x)$
• What does the first derivative of a function represent?

  Slope of the tangent line
• If f''(c) > 0</latex>, what type of relative extremum does $f(x)$ have at $x =$$c$?


  Local minimum
• If $f''(c) > 0$, then $f(x)$ has a local minimum at $x =$$c$


  True
• What does the Second Derivative Test use to identify extrema?

  Value of $f''(x)$
• Match the function with its derivative:

  $sin(x)$ ↔️ $cos(x)$

  $cos(x)$ ↔️ $- sin(x)$

  $e^{x}$ ↔️ $e^{x}$

  $ln(x)$ ↔️ $1 / x$
• Steps to find the second derivative of a function

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

  2️⃣ Differentiate $f'(x)$ to find $f''(x)$
• To find the critical points of a function, set the first derivative equal to zero
• How do you identify the critical points of a function $f(x)$?


  Solve $f'(x) =$$0$
• If $f''(c) < 0$, then f(x)</latex> has a local maximum
• The Second Derivative Test becomes inconclusive when f''(c) = 0
• What is the derivative of $sin(x)$?


  $cos(x)$
• The first derivative $f'(x)$ is derived from the original function $f(x)$.


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


  1️⃣ Compute the first derivative $f'(x)$.

  2️⃣ Set $f'(x) =$$0$.

  3️⃣ Solve for $x$ to find critical points.
• What do you set the first derivative equal to in order to find critical points?

  Zero
• The Second Derivative Test evaluates the second derivative at critical points.
• What is the value of $f''(1)$ for $f(x) =$$x^{3} - 6x^{2} +$$9x +$$1$?


  $- 6$
• If $f''(c) < 0$, then $f(x)$ has a local maximum at $x =$$c$.
• Steps to apply the Second Derivative Test

  1️⃣ Evaluate the second derivative $f''(x)$ at each critical point c</latex>.

  2️⃣ Check the sign of $f''(c)$ to determine the type of extrema.
• The second derivative describes the rate of change of the slope
• The second derivative is derived from the first derivative
• Critical points are the values of $x$ that satisfy the equation f'(x) = 0
• Steps to find critical points of a function $f(x)$


  1️⃣ Compute the first derivative $f'(x)$.

  2️⃣ Set $f'(x) =$$0$.

  3️⃣ Solve the equation $f'(x) =$$0$ for $x$.
• Under what condition is the Second Derivative Test inconclusive?

  $f''(c) =$$0$
• Match the critical point with the type of extrema based on the Second Derivative Test:

  $x =$$1$ ↔️ Local maximum

  $x =$$3$ ↔️ Local minimum
• Which test should you use if $f''(c) =$$0$?


  First Derivative Test
• The Second Derivative Test identifies extrema based on the sign of $f''(x)$ at critical points.
• The Second Derivative Test is used to determine whether a function has a local maximum or minimum