Re: Practice problems help:

#1 $\displaystyle \dfrac{384}{x} = 384x^{-1}$. You will eventually want to take a derivative, and it is much easier to avoid mistakes with the power rule than with the quotient rule.

$\displaystyle A(x) = (x+16)(384x^{-1}+18)$

Next, multiply it out. Do you remember the FOIL method? It tells you what to multiply. First Outside Inside Last (FOIL) means multiply the first terms: $\displaystyle x\cdot 384x^{-1}$, then the outside terms: $\displaystyle x\cdot 18$ then the inside terms: $\displaystyle 16\cdot 384x^{-1}$ then the last terms: $\displaystyle 16\cdot 18$. Add them all together. Then take the derivative and set it equal to zero. Next, take the second derivative and plug in any critical values you found. If the second derivative is positive at that value, you found a local minimum.

#2 I assume you are practicing integration. $\displaystyle \int f''(x)dx = f'(x) + C_1$ where $\displaystyle C_1$ is an arbitrary constant. Since you know $\displaystyle f'(0) = 3$, you can solve for $\displaystyle C_1$ after you integrate by plugging in $\displaystyle 0$ for $\displaystyle x$. Then, you integrate again: $\displaystyle \int f'(x)dx = f(x) + C_2$. You are given $\displaystyle f(0)=4$, so you can solve for $\displaystyle C_2$ after this second integration by plugging in $\displaystyle 0$ for $\displaystyle x$.

Note: If $\displaystyle \int f''(x)dx = f'(x)+C_1$ then $\displaystyle f'(x) = \int f''(x)dx - C_1$.