# Calculus  A particle moves along 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...

Calculus

A particle moves along 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. 1<-2, -1<t<1, and t>2

sciencesolve | Teacher | (Level 3) Educator Emeritus

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The positive values for x(t) lie on the right side, hence the negative values for x(t) express the movement to the left, thus you need to determine the intervals where the function `x(t)`  decreases.

You need to find derivative of function `x(t)=3t^5 - 25t^3+60t ` such that:

`x'(t)=15t^4 - 75t^2 + 60`

You need to solve the equation `x'(t)=0`  such that:

`15t^4 - 75t^2 + 60= 0`

You need to divide by 15 all over such that:

`t^4 - 5t^2 + 4 = 0`

`t^2=1 =gt t_(1,2)=+-1`

`t^2=4 =gt t_(3,4)=+-2`

The roots of equation are `t_1=+-1`  and `t_2=+-2` , hence you need to select a value for t between `+-1`  and `+- 2`  and you need to evaluate x'(t) at the value selected such that::

`t=0 =gt 0 - 0 + 4 = 4gt0`

You need to select a value for t smaller than -2 such that:

`x=-3 =gt 81-45+4 = 40gt0`

You need to select a value for t between -2 and -1 such that:

`t=-1.5 =gt 5.0625 - 11.25 + 4= -2.1875lt0`

You need to select a value for t between 1 and 2 such that:

`t=1.5 =gt 5.0625 - 11.25 + 4= -2.1875lt0`

Hence, the particle moves to the left if `t in (-2,-1)`  and `t in (1,2), ` thus you need to select B answer.