8. anyone know calculus?

Question

if a particle moves on a line according to the law s= t^5 + 2t^3, then the number of times it reverses direction is

Answer 1

The number of times it reverses direction
is the number of times the velocity
changes sign from+ to - or vice versa.
Now let's calculate v(t)
v(t)=s'(t)=5t^4+6t^2
v(t)=t^2*(5t^2+6)
because 5t^2+6>0
V(t)>=0 for t>:=0
So the particle doesn;t
reverses direction

Answer 2

The equation gives position as time changes. Find the derivative of the function to determine a function for its velocity.

v(t) = 5t^4 + 6t^2

Direction reverses whenever velocity changes sign. Another way of looking at the question is "how many maxima and minima are there?".

Solve for all points where v = 0. Thus: t^2(5t^2 + 6) = 0. There is one solution to the equation, but the velocity will never be negative since t^2 will remain positive. The particle does not change directions. This can be confirmed by graphing the position and velocity functions.

Answer 3

ds/dt gives the sign of velocity and so the direction of movement ds/dt =5t^4+6t^2 always >0 so it never reverses direction.
I suppose that it starts at t=0