Chapter 4.4, Problem 1E

To determine

To find: the local extrema, intervals of concave upward and concave downward, points

of inflection and then to sketch the graph of f.

Answer to Problem 1E

The function $f(x)$ has

local minima at $(1,1)$ and local maxima at $(1/3,31/27)$

The interval of concave upward is $(2/3,\infty )$ and concave downward is $(-\infty ,2/3)$

The inflection points at $x=2/3$ ,

Explanation of Solution

Given information:

  $f(x)=x^3-2x^2+x+1$

Formula used:

Second derivative test:

1. find $f\text{'}(x),f\text{'}\text{'}(x).$
2. find critical points by solving $f\text{'}(x)=0$
3. $f\text{'}\text{'}(c)<0,$ has local maximum $f\text{'}\text{'}(c)>0$ has local minimum at $x=c.$
4. for all x in the interval, $f\text{'}\text{'}(x)>0,$ then the function is concave upward $f\text{'}\text{'}(x)<0$,then the function is concave down ward.

Calculation:

Consider the function $f(x)=x^3-2x^2+x+1$

The domain of f is all real numbers.

Find the critical points

  $f\text{'}(x)=3x^2-4x+1$

  $f\text{'}(x)=3x^2-4x+1=0$

  $(x-1)(x-(1/3))=0$

Critical points are $x=1,1/3$

Find local minima and local maxima

  $f\text{'}\text{'}(x)=6x-4$

  $\begin{array}{l}x=1\Rightarrow \\ f\text{'}\text{'}(1)=6-4=2>0\text{ local minimum}\end{array}$

  $\begin{array}{l}x=1/3\Rightarrow \\ f\text{'}\text{'}(1/3)=6(1/3)-4=-2<0\text{ local maxima}\end{array}$

  $\begin{array}{l}f(1)=1\\ f(1/3)=31/27\end{array}$

Find the inflection points.

  $\begin{array}{l}f\text{'}\text{'}(x)=0\\ f\text{'}\text{'}(x)=6x-4=0\\ x=2/3\end{array}$

Test the intervals $(-\infty ,2/3),(2/3,\infty ).$

  $\begin{array}{l}(-\infty ,2/3)\Rightarrow \\ f\text{'}\text{'}(0)=6(0)-4=-4<0\text{ concave downward}\\ (2/3,\infty )\Rightarrow \\ f\text{'}\text{'}(1)=6(1)-4=2>0\text{ concave upward}\end{array}$

The inflection points at $x=2/3$

Graph:

The graph of $f(x)=x^3-2x^2+x+1$

  

Hence, local minima at $(1,1)$ and local maxima at $(1/3,31/27)$

The interval of concave upward is $(2/3,\infty )$ and concave downward is $(-\infty ,2/3)$

The inflection points at $x=2/3$ ,