A function $f\left(x\right)$ has the following properties:

$f^{\prime }\left(x\right)>0$ and $f^{\prime }$${}^{\prime }\left(x\right)>0$, for $x>4$

Which of the following is a possible graph of $f\left(x\right)$?

Graph the function $f\left(x\right)=6-6x^{\frac{2}{3}}+x^{\frac{8}{3}}$. The first and second derivative are given.

$f^{\prime }\left(x\right)=-\frac{4}{x^{\frac{1}{3}}}+\frac{8}{3}{x}^{\frac{5}{3}}$

$f^{\prime \prime }\left(x\right)=\frac{4}{3x^{\frac{4}{3}}}+\frac{40}{9}{x}^{\frac{2}{3}}$

Graph the given classical curve using analytical methods.

$y^2=x^3-2x+3$ ; $\frac{dy}{dx}=\frac{3x^2-2}{2y}$

In a computer simulation, a drone is programmed to always fly directly towards a moving target. The target moves north from a specific point at a constant speed of $1$ unit per time step. The drone starts its pursuit $3$ units east of the starting point of the target, moving at a speed of $s>1$ unit per time step. The path followed by the drone is given by the equation $y=\frac{s}{2}[\frac{x^{\frac{s+1}{s}}}{s+1}-\frac{x^{\frac{s-1}{s}}}{s-1}]+\frac{s}{s^2-1}$. For a simulation where $s=3$, what is the $y$-coordinate of the drone's position when $x=8$ units? Also, plot the graph for $s=3$.

The first derivative of a continuous function $g\left(x\right)$ is $g^{\prime }\left(x\right)=2x{(x-2)}^2$. Find the second derivative of $g\left(x\right)$ and sketch the general shape of its graph.