4.1: Applications of the First Derivative

Increasing/Decreasing Functions

A function is increasing on an interval $(a, b)$ if for any two numbers $x_1$ and $x_2$ in $(a, b)$, we have $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.
A function is decreasing on an interval $(a, b)$ if for any two numbers $x_1, x_2 \in (a, b)$, we have $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.
The derivative tells us exactly where a function is increasing and decreasing.
  1. If $f'(x) > 0$ for $x \in (a, b)$ then $f$ is increasing on $(a, b)$.
  2. If $f'(x) < 0$ for $x \in (a, b)$ then $f$ is decreasing on $(a, b)$.
Find intervals where $f(x) = x^2$ is increasing.
Determining Intervals Where a Function is Increasing or Decreasing
  1. Find all values of $x$ where $f'(x) = 0$ or $f'$ is discontinuous and identify the open intervals determined by these numbers.
  2. For each open interval, select a test number $c$ and determine the sign of $f'(c)$.
    1. If $f'(c) > 0$, then $f$ is increasing on that interval.
    2. If $f'(c) < 0$, then $f$ is decreasing on that interval.
Determine the intervals where $f(x) = x^3 - 3x^2 - 24x + 32$ is increasing/decreasing.
Determine the intervals where $f(x) = x^{2/3}$ is increasing/decreasing.
Determine the intervals where \[f(x) = x + \dfrac{1}{x}\] is increasing/decreasing.

Relative Extrema

Imagine climbing Madonna Mountain and standing on the peak. In terms of your elevation, two phenomena happen:

  1. You are higher than any point near you and
  2. even though Bishop's Peak is higher than you, if your context includes only Madonna Mountain, you are standing on the highest point.

These two ideas are the basis for the next definition:

A function $f$ has a relative maximum at $x=c$ if there exists an open interval $(a, b)$ containing $c$ such that $f(x) \leq f(c)$ for all $x \in (a,b)$.

Yellow text is phenomena two above while aqua text is condition one above.