My problem with the velocity of the object with position $z=t^2-t^3$.

Question

(a)

(b)Using first derivative test, $$\frac{dz}{dt}=0.$$ We get $t=2/3$. When $t=2/3$ object move $z=4/27$ units far from $z=0$.

(c) Velocity at which object departed from $z=0,$ $$\frac{dz}{dt}|_{t=0}=0.$$ Am I correct? What about the velocity at which it return? Is it $$\frac{dz}{dt}|_{t=1}?$$

(d)Is $z=f(t)$ unique?

May I know where is my mistake? Please help me to complete the answer.

Answer 1

You are OK so far and your graph works for the first 3 parts of the problem.

For part $(d)$ they want to find a different function with both velocities being zero at $t=0$ and $t=1$

You consider a polynomial in form of $bt^2+ct^3$ and find the coefficients. It should work out.