Answer
(a) $0$
(b) $112$
(c) $360$
Work Step by Step
(a) $f'(x) = 6x$
$\Rightarrow f''(x) = 6$
$\Rightarrow f'''(x) = 0$
$\Rightarrow f'''(2) = 0$
(b) $\frac{dy}{dx} = 30x^{4} - 8x$
$\Rightarrow \frac{d^{2}y}{dx^{2}} = 120x^{3} - 8$
$\Rightarrow \frac{d^{2}y}{dx^{2}}\vert_{x=1} = 120(1)^{3} - 8$
$= 120 - 8$
$=112$
(c) $\frac{d}{dx}[x^{-3}] = -3x^{-4}$
$\Rightarrow \frac{d^{2}}{dx^{2}}[x^{-3}] = 12x^{-5}$
$\Rightarrow \frac{d^{3}}{dx^{3}}[x^{-3}] = -60x^{-6}$
$\Rightarrow \frac{d^{4}}{dx^{4}}[x^{-3}] = 360x^{-7}$
$\Rightarrow \frac{d^{4}}{dx^{4}}[x^{-3}]\vert_{x=1} = 360(1)^{-7}$
$=360$
