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First we give a brief intuitive explanation:

Friction acts in the direction opposite motion both going up and coming down, dissipating some of the original KE.  So the automobile ends up moving more slowly at the end. 

Looking at the energy changes in more detail:

Intuitively:

• We can think about what happens in the process of going up and coming back down.  In the absence of friction the initial KE would convert to an equal amount of PE gain, all of which would then converted back to KE on the return. 

• With friction present, however, the automobile doesn't climb as high as it would if friction was not present.  There will be less PE to loses, and additional energy will be lost to friction on the way back down.

A little more rigorously:

• As the automobile coasts along the incline, in whichever direction, it will experience a force of rolling friction in the direction opposite its motion.  The force it exerts against friction is in the direction of its motion, so it does positive work. 

• The automobile will continue coasting up the incline until its KE becomes zero (i.e., until it comes to rest for an instant). 

• Since it does positive work against friction, its total change in PE as it travels up the incline will be less than its original KE. 

It will then coast back down the incline, losing PE and again doing work against friction. 

When it returns to its starting position, its loss of PE on the way down have been equal to the PE gain it experienced going up.  T

he PE loss on the way down is therefore less than its original KE, so even without the friction loss on the way down it would would end up with less KE than when it started. 

The frictional force on the way down once again opposes motion, so some of the PE loss is dissipated against friction, leaving it with even less KE. 

Now let's put this specifically into the context of the work-energy theorem, which can be stated in either of two equivalent ways:

• `dPE + `dKE + `dW_by_noncons = 0, where `dW_by_noncons is the work done BY the sytem (in this case the automobile) against nonconservative forces, or equivalently

• `dW_on_nc = `dKE + `dPE.  where `dW_on_nc is the work done ON the system by the nonconservative frictional force

We can use either formulation, depending on whether we want to think in terms of `dW_by_noncons (as in the first formulation) or `dW_on_noncons (as in the second formulation):

• If we want to view this situation from the point of view of the automobile, it is doing positive work against the frictional force so `dW_by_noncons is positive and this will lead us to the first formulation.

• If we want to view this from the point of the forces acting on the automobile, the frictional force exerted on it is doing negative work.  So `dW_on_noncons is negative and this will lead us to the second formulation.

To answer the question, it isn't necessary to consider anything but the fact that the automobile ends up where it started, and in the process does positive work against friction (or alternatively that friction does negative work on it):

Between the instant it starts up the incline and the instant it returns to its initial point, its PE change is zero, and it does positive work against the frictional force. 

• Using the formulation

`dPE + `dKE + `dW_by_noncons = 0

of the work-energy theorem, since `dW_by_noncons is positive and `dPE is zero we have

0 + `dKE + `dW_by_noncons = 0, which we rearrange into the form

`dKE = - `dW_by_noncons. 

The right-hand side is negative, so `dKE is negative.  The automobile ends up moving more slowly than when it started.

• Using alternative formulation of the work-energy theorem we have

`dW_on_nc = `dKE + `dPE. 

`dW_on_nc is negative, and `dPE is again zero, so 

`dKE = `dW_on_nc,

which is negative. 

We again conclude that the automobile ends up moving more slowly than when it started.