### Home > APCALC > Chapter 12 > Lesson 12.2.1 > Problem12-71

12-71.

The function $f$ has derivatives of all orders within its radius of convergence of $\frac { 1 } { 3 }$. Its Maclaurin series is $p ( x ) = \displaystyle \sum _ { n = 0 } ^ { \infty } 3 ^ { n } x ^ { n + 2 }$.

1. What are the coefficients of the first-degree and second-degree terms of $p(x)$? Use those coefficients to determine if $f$ has a local maximum, local minimum or neither at $x = 0$. Justify your answer.

$p(x)=3^0x^{0+2}+3^1x^{1+2}+3^2x^{2+2}+3^3x^{3+2}+...$

$p^\prime (0) = ?$

$p^{\prime \prime}(0) = ?$

2. Expand the Maclaurin series for $f$ out to four terms to create a fifth-degree Taylor polynomial centered at $x = 0$, $p_5(x)$. Then write the antiderivative of $p_5(x)$.

$\int(x^2+3x^{3}+9x^{4})dx$