# Speed Question

Given the function $f(x)=x^3 -4x+2$ let f represent the position of an object with respect to time that is moving along a line. Identify when the object is moving in the positive direction, when it is moving in the negative direction and when it is at rest. Show all work.

f(x) is the position of the object with respect to time denoted by the variable x.

Hence the speed of the object is $speed = df/dx$ equal to the first derivative of the posion in the variable time (x here). the object is moving with positive speed when the first derivative is positive, with negative speed when the first derivative is negative and is at rest when the first derivative is zero.

$df/dx = 3*x^2 -4$

$3x^2 -4 =0$, $x1=\sqrt{4/3}$, $x2=-\sqrt{-4/3}$,

It means the object is at rest at the time $x = \sqrt{4/3}$, is moving with positive speed (toward positive direction) when x greater then $\sqrt{4/3}$ and is moving with negative speed when x is lesser than $\sqrt{4/3}$. Negative values of x does not carry any physical meaning since negative time does not exist.