Answer:
The maximum acceleration over that interval is  .
. 
Step-by-step explanation:
The acceleration of this car is modelled as a function of the variable  .
.
Notice that the interval of interest  is closed on both ends. In other words, this interval includes both endpoints:
 is closed on both ends. In other words, this interval includes both endpoints:  and
 and  . Over this interval, the value of
. Over this interval, the value of  might be maximized when
 might be maximized when  is at the following:
 is at the following:
- One of the two endpoints of this interval, where  or or . .
- A local maximum of  , where , where (first derivative of (first derivative of is zero) and is zero) and (second derivative of (second derivative of is smaller than zero.) is smaller than zero.)
Start by calculating the value of  at the two endpoints:
 at the two endpoints: 
 . .
 . .
Apply the power rule to find the first and second derivatives of  :
:
 .
.
 .
.
Notice that both  and
 and  are first derivatives of
 are first derivatives of  over the interval
 over the interval  .
. 
However, among these two zeros, only  ensures that the second derivative
 ensures that the second derivative  is smaller than zero (that is:
 is smaller than zero (that is:  .) If the second derivative
.) If the second derivative  is non-negative, that zero of
 is non-negative, that zero of  would either be an inflection point (if
 would either be an inflection point (if ) or a local minimum (if
) or a local minimum (if  .)
.) 
Therefore  would be the only local maximum over the interval
 would be the only local maximum over the interval  .
.
Calculate the value of  at this local maximum:
 at this local maximum:
 . .
Compare these three possible maximum values of  over the interval
 over the interval  . Apparently,
. Apparently,  would maximize the value of
 would maximize the value of  . That is:
. That is:  gives the maximum value of
 gives the maximum value of  over the interval
 over the interval  .
. 
However, note that the maximum over this interval exists because  is indeed part of the
 is indeed part of the  interval. For example, the same
 interval. For example, the same  would have no maximum over the interval
 would have no maximum over the interval  (which does not include
 (which does not include  .)
.)