(DG2) Computationally Finding Intervals of Increase / Decrease#

By the end of the lesson you will be able to:

  • locate the intervals of increase and the intervals of decrease using the first derivative of a function.

Lecture Videos#

Finding Intervals of Increase / Decrease#

Increase / Decrease Test

For the interval \((a,b)\), if \(f\) is differentiable then:

  • \(f'(x)>0 \iff f\) is increasing

  • \(f'(x)<0 \iff f\) is decreasing

Critical Number

The number \(x=c\) is a critical number of \(f\) provided either:

\[ f'(c)=0 \qquad \text{or} \qquad f'(c) \;\text{ DNE} \]

and \(c\) is in the domain of \(f\).

Strategy

  1. Calculate \(f'(x)\)

  2. Find where \(f'(x)=0\) and \(f'(x) \;\text{ DNE}\).

  3. Create a sign chart for \(f'\).

  4. Determine the sign of \(f'\) on each interval.

Example 1#

Find the intervals of increase and the intervals of decrease for function \(f\) with first derivative given below:

\[ f'(x)= x(x-1)^2(x+3)^{-3} \]

Example 2#

Find the intervals of increase and the intervals of decrease for function \(f\) given below:

\[ f(x)= \dfrac{1}{3}x^3-2x^2+3x+1 \]

Example 3#

Find the intervals of increase and the intervals of decrease for function \(f\) given below:

\[ f(x)= x^4-4x^3 \]

Example 4#

Find the intervals of increase and the intervals of decrease for function \(f\) given below:

\[ f(x)= \dfrac{2x^2}{x^2-1} \]

Example 5#

Find the intervals of increase and the intervals of decrease for function \(f\) given below:

\[ f(x)= x + \sqrt{x} \sin x \qquad \text{with} \qquad 0\leq x \leq 2\pi \]