Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 257: 66

$$\eqalign{ & {\text{Concave up on }}\left( { - \infty ,0} \right) \cr & {\text{Concave down on }}\left( {0,\infty } \right) \cr & {\text{The inflection points is at: }}x = 0 \cr & \cr} $$

$$\eqalign{ & f\left( x \right) = {\tan ^{ - 1}}x \cr & {\text{Calculate the second derivative}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\tan }^{ - 1}}x} \right] \cr & f'\left( x \right) = \frac{1}{{1 + {x^2}}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{1}{{1 + {x^2}}}} \right] \cr & f''\left( x \right) = - \frac{{2x}}{{{{\left( {1 + {x^2}} \right)}^2}}} \cr & \cr & {\text{Set the second derivative to 0}} \cr & - \frac{{2x}}{{{{\left( {1 + {x^2}} \right)}^2}}} = 0 \cr & 2x = 0 \cr & {\text{We see that }}\,f''\left( x \right) = 0{\text{ at }}x = 0 \cr & {\text{That points is candidate for inflection points}} \cr & {\text{We need evaluate the intervals }}\left( { - \infty ,0} \right){\text{ and }}\left( {0,\infty } \right) \cr & {\text{Now}}{\text{, we will evaluate test values and resume in a table}} \cr & {\text{to determinate whether the concavity changes at these points}} \cr} $$ \[\begin{array}{*{20}{c}} {{\text{Interval}}}&{{\text{Test value }}\left( x \right)}&{{\text{Sign of }}f''\left( x \right)}&{{\text{Behavior of }}f\left( x \right)} \\ {\left( { - \infty ,0} \right)}&{ - 1}&{f''\left( { - 1} \right) > 0}&{{\text{Concave up}}} \\ {\left( {0,\infty } \right)}&1&{f''\left( 1 \right) < 0}&{{\text{Concave down}}} \end{array}\] $$\eqalign{ & {\text{From the table we can conclude that the function is:}} \cr & {\text{Concave up on }}\left( { - \infty ,0} \right) \cr & {\text{Concave down on }}\left( {0,\infty } \right) \cr & {\text{The inflection points is at: }}x = 0 \cr} $$