Solution: $P(t)=12000\sqrt[\phantom{2}]{t+4}-23500.$

$2.(20$ points$)$ Find a function $f(x)$ such that $\frac{d^{2}f}{d{x}^2}=\frac{1}{\sqrt[\phantom{2}]{x}}+e^{-2x}+3,$ subject to $f^{\text{'}}(0)=\frac{5}{2},$ and $f(0)=\frac{3}{4}.$ Solution: $f(x)=\frac{3}{2}{x}^2+\frac{4}{3}{x}^{\frac{3}{2}}+3x+\frac{1}{4}{e}^{-2x}+\frac{1}{2}.$