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How do you solve this multiple choice calculus problem?

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


Is choice "B" supposed to be -2<t<-1 and 1<t<2..????
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2 Answers

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.
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)