Applications of the First Derivative

Applications of the First Derivative Increasing/Decreasing Theorem

1. Theorems regarding the First Derivative

1.1 Conditions for Increasing/Decreasing Functions

• If f′(x) > 0 for every value of x in an interval (a, b):

  • Then f is increasing on (a, b).

• If f′(x) < 0 for every value of x in an interval (a, b):

  • Then f is decreasing on (a, b).

• If f′(x) = 0 for every value of x in an interval (a, b):

  • Then f is constant on (a, b).

1.2 Summary

• The first derivative f′ determines whether f(x) is increasing or decreasing in the specified interval.

2. Determining the Intervals Where a Function is Increasing or Decreasing

2.1 Steps to Find Intervals

1. Find all values of x for which f′(x) = 0 or f′(x) is undefined.

   • These are x-values where f′(x) changes.

2. Perform a sign analysis.

   • Divide the real number line by the numbers found in Step 1.

   • Select a convenient number in each interval from step 1, and determine the sign of f′ in that interval.

   • If f′(x) > 0, f is increasing on that interval.

   • If f′(x) < 0, f is decreasing on that interval.

2.2 Example 1a

• Given the function: $f(x) = -x^3 + 6x^2$

  • Task: Find the intervals where f(x) is increasing and where f(x) is decreasing.

Applications of the First Derivative Relative Extrema Definitions

3. Critical Points and Extrema

3.1 Definitions

• A function f has a relative maximum at x = c if:

  • There exists an open interval (a, b) containing c such that f(c) ≥ f(x) for all x in (a, b).

• A function f has a relative minimum at x = c if:

  • There exists an open interval (a, b) containing c such that f(c) ≤ f(x) for all x in (a, b).

• A critical number of a function f is any number x in the domain of f (i.e. where f is defined) such that f′(x) = 0 or f′(x) does not exist.

  • For continuous functions, the collection of critical numbers encompasses all relative extrema.

3.2 The First Derivative Test

3.2.1 Procedure for Finding Relative Extrema

1. Determine the critical numbers of f.

2. Determine the sign of f′(x) to the left and right of each critical number:

   • If f′(x) changes sign from positive to negative as we move across a critical number c, then f has a relative maximum at x = c.

   • If f′(x) changes sign from negative to positive as we move across a critical number c, then f has a relative minimum at x = c.

   • If f′(x) does not change sign as we move across a critical number c, then f has no relative extrema at x = c.

3.3 Example 1b

• Consider the function: $f(x) = -x^3 + 6x^2$

  • Task: Find the relative maxima and relative minima.

Additional Examples

4. Further Applications of the First Derivative

4.1 Example 2

• Given the function: $g(t) = 1 - t(t^2)$

  • Tasks:

  • (a) Find the interval(s) where g(t) is increasing and the intervals where g(t) is decreasing.

  • (b) Find the relative maxima and relative minima of g(t), if they exist.

4.2 Example 3

• Given the function: $h(x) = rac{x + 1}{x}$

  • Tasks:

  • (a) Find the interval(s) where h(x) is increasing and the intervals where h(x) is decreasing.

  • (b) Find the relative maxima and relative minima of h(x), if they exist.

4.3 Example 4

• Given the function: $f(x) = rac{1}{3}x^3 - 3x^2 + 9x$

  • Task: Find all relative maxima and relative minima, if they exist.

  • This example illustrates that a critical number does not guarantee a relative max or min.

4.4 Example 5

• Given the provided graph of f:

  • Tasks:

  • (a) What are the critical numbers of f?

  • (b) Draw the sign diagram for f′.

  • (c) Find the relative extrema of f.

4.5 Example 6

• Given the graph of the derivative f′:

  • Task: Determine where the graph of f is increasing, constant, and decreasing.