How do you solve this multiple choice calculus problem?

Question

A particle moves on the x-axis in such a way that its position at time t is given by x(t)=3t⁵ - 25t³ +60t.

For what values of t is the particle moving to the left?

A) -2<t<1 only

B) -2<t<1 and 1<t<2

C) -1<t<1 and t>2

D) 1<t<2 only

E) t<-2, -1<t<1, and t>2

Answer 1

1. A particle is moving to the left when its velocity is negative.

2.  The velocity is the derivative of the displacement versus time function.

 

v(t) = s'(t) =15t⁴-75t²+60 <= 0

 

This function can be factored by first factoring out the 15 to give 15(t⁴-5t²+4) which in turn can be factored

15(t² - 4)(t² - 1).   The two terms can be further factored because each is the difference of two squares.

 

15(t-2)(t+2)(t-1)(t+1) < 0.    Kenneth has provided the rest of the solution.  I just wanted to help you factor the polynomial.

 

 

Answer 2

The correct answer is E because x'(t) = 15(t-2)(t+2)(t-1)(t+1)

and this velocity polynomial is positive on three intervals: (-infinity,-2) (-1,1)  (2,infinity)