5.4 Using the First Derivative Test for Relative Extrema

• What are relative extrema on a function's graph?

  Local maxima or minima
• The first derivative test identifies relative extrema by examining the sign change of the derivative around a critical
• A local minimum occurs when the function changes from decreasing to increasing.
• Steps of the First Derivative Test

  1️⃣ Find the critical points where $f'(x) =$$0$ or $f'(x)$ is undefined

  2️⃣ Create a sign chart using critical points

  3️⃣ Test intervals in the sign chart to determine $f'(x)$ sign

  4️⃣ Determine relative extrema based on sign changes
• If $f'(x)$ changes from positive to negative at a critical point, it indicates a local maximum
• If $f'(x)$ changes from negative to positive at a critical point, it's a local minimum.
• What type of extrema does f(x) = x^{3} - 3x</latex> have at $x =$$- 1$?


  Local maximum
• To find critical points, set the derivative $f'(x)$ equal to zero
• What is the critical point of $f(x) =$$x^{2} - 4x +$$5$?


  $x =$$2$
• Match the concept with its definition:

  Relative extrema ↔️ Local maxima or minima

  First Derivative Test ↔️ Identifies relative extrema

  Critical points ↔️ Where $f'(x) =$$0$ or $f'(x)$ is undefined
• What are relative extrema on a function's graph?

  Local maxima or minima
• The first derivative test helps identify relative extrema by examining the sign change of the derivative
• What is the derivative of $f(x) =$$x^{3} - 3x$?


  $f'(x) =$$3x^{2} - 3$
• The function $f(x) =$$x^{3} - 3x$ has relative extrema at $x =$$- 1$ and $x =$$1$
• What is a local maximum in terms of function behavior?

  Increasing to decreasing
• Summarize the function behavior at a local minimum.

  Decreasing to increasing
• At a local minimum, the derivative sign changes from negative
• What are the critical points in the first derivative test?

  Where f'(x) = 0
• In the first derivative test, a sign chart is created to divide the number line based on critical points.
• Steps to apply the first derivative test to $f(x) =$$x^{3} - 3x$


  1️⃣ Find the derivative $f'(x) =$$3x^{2} - 3$

  2️⃣ Set $f'(x) =$$0$ and solve for $x$

  3️⃣ Create a sign chart with critical points $x =$$- 1$ and $x =$$1$

  4️⃣ Test intervals: $( - \infty, - 1)$, $( - 1, 1)$, $(1, \infty)$

  5️⃣ Determine function behavior: increasing, decreasing, increasing

  6️⃣ Conclude that there is a local maximum at $x =$$- 1$ and a local minimum at $x =$$1$
• If $f'(x)$ changes from positive to negative at a critical point, it is a local maximum
• If $f'(x)$ does not change sign at a critical point, there is no relative extremum.
• What is the derivative of $f(x) =$$x^{3} - 3x$?


  $f'(x) =$$3x^{2} - 3$
• The critical points of $f(x) =$$x^{3} - 3x$ are $x =$$- 1$ and $x =$$1$ because $f'(x) =$$0$ at these values
• The function $f(x) =$$x^{3} - 3x$ has a local maximum at $x =$$- 1$
• What is the purpose of the First Derivative Test?

  Find relative extrema
• Steps of the First Derivative Test

  1️⃣ Find the critical points

  2️⃣ Create a sign chart

  3️⃣ Test intervals

  4️⃣ Determine extrema
• If $f'(x)$ changes from negative to positive at a critical point, it is a local minimum
• To find critical points, you must set $f'(x) =$$0$ and solve for $x$.
• What is the critical point of $f(x) =$$x^{2} - 4x +$$5$?


  $x =$$2$
• If $f'(x) > 0$ in an interval, then $f(x)$ is increasing in that interval.
• What are the critical points of $f(x) =$$x^{3} - 3x^{2} +$$1$?


  $x =$$0$ and $x =$$2$
• If $f'(x)$ changes from positive to negative at a critical point, there is a local maximum
• The function $f(x) =$$x^{3} - 3x^{2} +$$1$ has a local minimum at $x =$$2$