Chapter 3 - Applications of Differentiation - 3.4 Exercises - Page 192: 20

$$\eqalign{ & {\text{Inflection points: }}\left( {\frac{3}{2}, - \frac{1}{{16}}} \right){\text{ and }}\left( {2,0} \right) \cr & {\text{Concave downward}}:{\text{ }}\left( {\frac{3}{2},2} \right) \cr & {\text{Concave upward}}:{\text{ }}\left( { - \infty ,\frac{3}{2}} \right){\text{ and }}\left( {2,\infty } \right) \cr} $$

$$\eqalign{ & f\left( x \right) = {\left( {x - 2} \right)^3}\left( {x - 1} \right) \cr & {\text{Calculate the second derivative}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {x - 2} \right)}^3}\left( {x - 1} \right)} \right] \cr & f'\left( x \right) = {\left( {x - 2} \right)^3}\frac{d}{{dx}}\left[ {\left( {x - 1} \right)} \right] + \left( {x - 1} \right)\frac{d}{{dx}}\left[ {{{\left( {x - 2} \right)}^3}} \right] \cr & f'\left( x \right) = {\left( {x - 2} \right)^3} + 3{\left( {x - 2} \right)^2}\left( {x - 1} \right) \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {x - 2} \right)}^3} + 3{{\left( {x - 2} \right)}^2}\left( {x - 1} \right)} \right] \cr & f''\left( x \right) = 3{\left( {x - 2} \right)^2} + 3{\left( {x - 2} \right)^2}\left( 1 \right) + 6\left( {x - 2} \right)\left( {x - 1} \right) \cr & f''\left( x \right) = 6{\left( {x - 2} \right)^2} + 6\left( {x - 2} \right)\left( {x - 1} \right) \cr & {\text{Set }}f''\left( x \right) = 0 \cr & 6{\left( {x - 2} \right)^2} + 6\left( {x - 2} \right)\left( {x - 1} \right) = 0 \cr & {\text{Factoring}} \cr & 6\left( {x - 2} \right)\left[ {\left( {x - 2} \right) + \left( {x - 1} \right)} \right] = 0 \cr & 6\left( {x - 2} \right)\left( {2x - 3} \right) = 0 \cr & x - 2 = 0,{\text{ }}2x - 3 = 0 \cr & x = 2,{\text{ }}x = \frac{3}{2} \cr & {\text{Set the intervals }}\left( { - \infty ,\frac{3}{2}} \right),\left( {\frac{3}{2},2} \right),\left( {2,\infty } \right) \cr & {\text{Making a table of values }}\left( {{\text{See examples on page 188 }}} \right) \cr} $$ \[\boxed{\begin{array}{*{20}{c}} {{\text{Interval}}}&{\left( { - \infty ,\frac{3}{2}} \right)}&{\left( {\frac{3}{2},2} \right)}&{\left( {2,\infty } \right)} \\ {{\text{Test Value}}}&{x = 0}&{x = 1.6}&{x = 3} \\ {{\text{Sign of }}f''\left( x \right)}&{36 > 0}&{ - \frac{{12}}{{25}} < 0}&{18 > 0} \\ {{\text{Conclusion}}}&{{\text{C}}{\text{. upward}}}&{{\text{C}}{\text{. downward}}}&{{\text{C}}{\text{. upward}}} \end{array}}\] $$\eqalign{ & {\text{The inflection points occur at }}x = \frac{3}{2}{\text{ and }}x = 2 \cr & f\left( {\frac{3}{2}} \right) = {\left( {\frac{3}{2} - 2} \right)^3}\left( {\frac{3}{2} - 1} \right) = - \frac{1}{{16}} \to \left( {\frac{3}{2}, - \frac{1}{{16}}} \right) \cr & f\left( 2 \right) = {\left( {2 - 2} \right)^3}\left( {2 - 1} \right) = 0 \to \left( {2,0} \right) \cr & {\text{Inflection points: }}\left( {\frac{3}{2}, - \frac{1}{{16}}} \right){\text{ and }}\left( {2,0} \right) \cr & {\text{Concave downward}}:{\text{ }}\left( {\frac{3}{2},2} \right) \cr & {\text{Concave upward}}:{\text{ }}\left( { - \infty ,\frac{3}{2}} \right){\text{ and }}\left( {2,\infty } \right) \cr} $$