Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 257: 64

$$\eqalign{ & {\text{Concave up on }}\left( { - \infty ,4} \right) \cr & {\text{Concave down on }}\left( {4,\infty } \right) \cr & {\text{The inflection points is at: }}x = 4 \cr} $$

$$\eqalign{ & g\left( x \right) = \root 3 \of {x - 4} \cr & {\text{Calculate the second derivative}} \cr & g'\left( x \right) = \frac{d}{{dx}}\left[ {\root 3 \of {x - 4} } \right] \cr & g'\left( x \right) = \frac{1}{3}{\left( {x - 4} \right)^{ - 2/3}} \cr & g''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{1}{3}{{\left( {x - 4} \right)}^{ - 2/3}}} \right] \cr & g''\left( x \right) = - \frac{2}{9}{\left( {x - 4} \right)^{ - 5/3}} \cr & g''\left( x \right) = - \frac{2}{{9{{\left( {x - 4} \right)}^{5/3}}}} \cr & {\text{The second derivative is not defined at }}x = 4 \cr & {\text{That points is candidate for the inflection points}} \cr & {\text{We need evaluate the intervals }}\left( { - \infty ,4} \right){\text{ and }}\left( {4,\infty } \right) \cr & {\text{Now}}{\text{, we will evaluate test values and resume in a table}} \cr & {\text{to determinate whether the concavity changes at these points}} \cr} $$ \[\begin{array}{*{20}{c}} {{\text{Interval}}}&{{\text{Test value }}\left( x \right)}&{{\text{Sign of }}g''\left( x \right)}&{{\text{Behavior of }}g\left( x \right)} \\ {\left( { - \infty ,4} \right)}&3&{g''\left( 3 \right) > 0}&{{\text{Concave up}}} \\ {\left( {4,\infty } \right)}&5&{g''\left( 5 \right) < 0}&{{\text{Concave down}}} \end{array}\] $$\eqalign{ & {\text{From the table we can conclude that the function is:}} \cr & {\text{Concave up on }}\left( { - \infty ,4} \right) \cr & {\text{Concave down on }}\left( {4,\infty } \right) \cr & {\text{The inflection points is at: }}x = 4 \cr} $$