Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 257: 58

$$\eqalign{ & {\text{Concave down on }}\left( { - \infty ,2} \right){\text{ and }}\left( {1,\infty } \right) \cr & {\text{Concave up on }}\left( { - 2,1} \right) \cr & {\text{The inflection points are at: }}x = - 2{\text{ and }}x = 1 \cr} $$

$$\eqalign{ & f\left( x \right) = - {x^4} - 2{x^3} + 12{x^2} \cr & \cr & {\text{Calculate the second derivative}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ { - {x^4} - 2{x^3} + 12{x^2}} \right] \cr & f'\left( x \right) = - 4{x^3} - 6{x^2} + 24x \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ { - 4{x^3} - 6{x^2} + 24x} \right] \cr & f''\left( x \right) = - 12{x^2} - 12x + 24 \cr & \cr & {\text{Set the second derivative to 0}} \cr & f''\left( x \right) = 0 \cr & - 12{x^2} - 12x + 24 = 0 \cr & {x^2} + x - 2 = 0 \cr & \left( {x + 2} \right)\left( {x - 1} \right) = 0 \cr & \cr & {\text{We see that }}\,f''\left( x \right) = 0{\text{ at }}x = - 2{\text{ and}}\,\,\,x = 1 \cr & {\text{These points are candidates for inflection points}} \cr & {\text{We need evaluate the intervals }}\left( { - \infty , - 2} \right),\,\,\left( { - 2,1} \right){\text{ and }}\left( {1,\infty } \right) \cr & \cr & {\text{Now}}{\text{, we will evaluate test values and resume in a table}} \cr & {\text{to determinate whether the concavity changes at these points}} \cr} $$ \[\begin{array}{*{20}{c}} {{\text{Interval}}}&{{\text{Test value }}\left( x \right)}&{{\text{Sign of }}f''\left( x \right)}&{{\text{Behavior of }}f\left( x \right)} \\ {\left( { - \infty , - 2} \right)}&{ - 3}&{f''\left( { - 3} \right) < 0}&{{\text{Concave down}}} \\ {\left( { - 2,1} \right)}&0&{f''\left( 0 \right) > 0}&{{\text{Concave up}}} \\ {\left( {1,\infty } \right)}&2&{f''\left( 2 \right) < 0}&{{\text{Concave down}}} \end{array}\] $$\eqalign{ & {\text{From the table we can conclude that the function is:}} \cr & {\text{Concave down on }}\left( { - \infty ,2} \right){\text{ and }}\left( {1,\infty } \right) \cr & {\text{Concave up on }}\left( { - 2,1} \right) \cr & {\text{The inflection points are at: }}x = - 2{\text{ and }}x = 1 \cr} $$