5.3 Determining Intervals on Which a Function is Increasing or Decreasing

• Critical points are found by setting the first derivative $f'(x)$ equal to zero
• If $f'(x) > 0$, the function is increasing.
• What is the first derivative of $f(x) =$$x^{2} - 4x +$$3$?


  $f'(x) =$$2x - 4$
• For the interval $( - \infty, 2)$, the value of $f'(x)$ is negative
• The function $f(x) =$$x^{2} - 4x +$$3$ is decreasing on $( - \infty, 2)$.
• Match the function property with its corresponding behavior:

  $f'(x) > 0$ ↔️ Increasing

  $f'(x) < 0$ ↔️ Decreasing
• A function is increasing if its value increases as the input variable increases
• A function is increasing if $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$, which means its slope is positive
• The function $f(x) =$$x^{2}$ is decreasing on ( - \infty, 0)</latex> and increasing on $(0, \infty)$.
• Order the 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 Check for values where $f'(x)$ is undefined
• What is a critical point of a function $f(x)$?


  $f'(c) =$$0$
• A sign chart helps determine intervals where a function is increasing or decreasing based on the sign of its first derivative
• If $f'(x) < 0$ on an interval, the function is decreasing on that interval.
• What is the first derivative of $f(x) =$$x^{3} - 6x^{2} +$$9x$?


  $3x^{2} - 12x +$$9$
• The critical points of $f(x) =$$x^{3} - 6x^{2} +$$9x$ are $x =$$1$ and x = 3
• Match the sign of $f'(x)$ with the behavior of the function.


  Positive ↔️ Increasing

  Negative ↔️ Decreasing
• What is the definition of an increasing function?

  $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$
• A function $f(x)$ is decreasing on an interval if $f(x_{1}) > f(x_{2})$ when $x_{1} < x_{2}$.decreasing
• What does the first derivative of a function determine?

  Whether it is increasing
• The first derivative $f'(x)$ is positive when a function is decreasing.


  False
• What does $f'(x) > 0$ indicate about the function $f(x)$?


  It is increasing
• If $f'(x) < 0$, the function $f(x)$ is decreasing
• Match the function type with its definition and first derivative:

  Increasing ↔️ $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$

  Decreasing ↔️ $f(x_{1}) > f(x_{2})$ when $x_{1} < x_{2}$
• Order the steps to determine whether a function is increasing or decreasing using its first derivative:

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

  2️⃣ Set $f'(x) > 0$ to find where the function is increasing

  3️⃣ Set $f'(x) < 0$ to find where the function is decreasing
• If $f'(x) > 0$, the function is increasing and $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$.
• A function $f(x)$ is increasing on an interval if $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$. Conversely, it is decreasing if $f(x_{1}) > f(x_{2})$ when $x_{1} < x_{2}$.decreasing
• If $f'(x) > 0$, the function is increasing.
• What determines whether a function is increasing or decreasing?

  First derivative
• If $f'(x) > 0$, the function is increasing
• A function $f(x)$ is decreasing on an interval if $f(x_{1}) > f(x_{2})$ when $x_{1} < x_{2}$. Conversely, it is increasing if $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$.increasing
• If $f'(x) < 0$, the function is decreasing.
• A function $f(x)$ is increasing on an interval if $f(x_{1}) < f(x_{2})$ when $x_{1} < x_{2}$. Conversely, it is decreasing if $f(x_{1}) > f(x_{2})$ when $x_{1} < x_{2}$.decreasing
• What is the condition for a function to be increasing based on its first derivative?

  $f'(x) > 0$
• Iff'(x) = 0</latex>, the function is neither increasing nor decreasing.
• On what interval is $f(x) =$$x^{2}$ decreasing?


  $( - \infty, 0)$
• The function $f(x) =$$x^{2}$ is decreasing on $( - \infty, 0)$ and increasing on $(0, \infty)$.increasing
• If $f'(x) > 0$, the function is increasing.
• On what interval is $f(x) =$$x^{2}$ increasing?


  $(0, \infty)$
• Order the steps to find the first derivative of a function.

  1️⃣ Apply derivative rules such as the Power Rule or Sum Rule.

  2️⃣ Simplify the result if necessary.
• What is the Power Rule for differentiation?

  $\frac{d}{dx}(x^{n}) =$$nx^{n - 1}$