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The logarithmic expression is
To expand the expression we have to use some properties of logarithm.
We know that
By using this property we can write,
Square root means to the power 1/2, so for , we can write
.
Now we have to use another property of logarithm.
We know that,
So we will use this property to
Now we have to use another property of logarithm.
We know that,
By using this property we can write,
This is the required aswer here.
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:
and
. Over this interval, the value of
might be maximized when
is at the following:
- One of the two endpoints of this interval, where
or
.
- A local maximum of
, where
(first derivative of
is zero) and
(second derivative of
is smaller than zero.)
Start by calculating the value of at the two endpoints:
.
.
Apply the power rule to find the first and second derivatives of :
.
.
Notice that both and
are first derivatives of
over the interval
.
However, among these two zeros, only ensures that the second derivative
is smaller than zero (that is:
.) If the second derivative
is non-negative, that zero of
would either be an inflection point (if
) or a local minimum (if
.)
Therefore would be the only local maximum over the interval
.
Calculate the value of at this local maximum:
.
Compare these three possible maximum values of over the interval
. Apparently,
would maximize the value of
. That is:
gives the maximum value of
over the interval
.
However, note that the maximum over this interval exists because is indeed part of the
interval. For example, the same
would have no maximum over the interval
(which does not include
.)