(DG4) Computationally Finding Intervals of Concavity#

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

  • locate the intervals of concavity using the second derivative.

  • locate the inflection points using the second derivative.

Lecture Videos#

Finding Intervals of Concavity#

Concavity and Second Derivatives

For the interval \((a,b)\):

  • \(f''(x)>0 \implies f\) is concave up

  • \(f''(x)<0 \implies f\) is concave down

Possible Inflection Number

The number \(x=c\) is a possible inflection number of \(f\) provided:

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

Finding Inflection Points#

Inflection Number

An inflection number of \(f\) is a number \(x=c\) in the domain of \(f\) provided:

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

and \(f''\) switches signs at \(x=c\).

Strategy

  1. Find the intervals of concavity.

  2. Determine the inflection numbers of \(f\) (check domain and change in concavity)

  3. Determine the \(y\)-coordinate for each inflection point.

Example 1#

Find the intervals of concavity for function \(f\) with second derivative given below:

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

Example 2#

Find the intervals of concavity and any inflection points for:

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

Example 3#

Find the intervals of concavity and any inflection points for:

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

Example 4#

Find the intervals of concavity and any inflection points for:

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

Example 5#

Find the intervals of concavity and any inflection points for:

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