(5.3) For each of the following functions identify all intervals where the graph of \( f \) is concave up and all intervals where the graph is concave down. Also find the \( x \) and \( y \) coordinates of any inflection points. (a) \( f(x)=x^{3}-6 x^{2}-15 x+7 \)

\( \lim _ { n \rightarrow \infty } [ \sum _ { i = 1 } ^ { n } \frac { 3 - ( - 1 ) } { n } \cdot f ( ( - 1 ) + i \cdot \frac { 3 - ( - 1 ) } { n } ) ] \)

\( 1 \leftarrow \) The brain mass of a fetus can be estimated using the total mass of the fetus by the function \( \mathrm{b}=0.22 \mathrm{~m}^{0.87} \), where m is the mass of the fetus (in grams) and b is the brain mass (in grams). Suppose the brain mass of a 27 - g fetus is changing at a rate of 0.21 g per day. Use this to estimate the rate of change of the total mass of the fetus, \( \frac{\mathrm{dm}}{\mathrm{dt}} \). \( \frac{\mathrm{dm}}{\mathrm{dt}}=\square \mathrm{g} \) per day (Round to five decimal places as needed.)

\( \lim _ { n \rightarrow \infty } [ \sum _ { i = 1 } ^ { n } \frac { 3 - ( - 1 ) } { n } \cdot f ( - 1 ) + i \cdot \frac { - ( - 1 ) } { n } ) ] \)

The brain mass of a fetus can be estimated using the total mass of the fetus by the function \( b=0.22 \mathrm{~m}^{0.87} \), where m is the mass of the fetus (in grams) and b is the brain ma: grams). Suppose the brain mass of a \( 21-\mathrm{g} \) fetus is changing at a rate of 0.26 g per day. Use this to estimate the rate of change of the total mass of the fetus, \( \frac{\mathrm{dm}}{\mathrm{dt}} \). \( \frac{\mathrm{dm}}{\mathrm{dt}}=\square \mathrm{g} \) per day (Round to five decimal places as needed.)

\( \int _ { - 1 } f ( x ^ { 2 } - 1 ) d x = \lim _ { n \rightarrow \infty } [ \sum _ { i = 1 } ^ { n } \frac { 3 - ( - 1 ) } { n } \cdot f ( ( - 1 ) + i \cdot \frac { 3 - ( - 1 ) } { n } ) ] \)