Question
The gas mileage M (s), in miles per gallon, of a car traveling s miles per hour is modeled by the function below, where 
.
According to the model, at what speed, in miles per hour, does the car obtain its greatest gas mileage?
- 46
- 48
- 50
- 75
Hint:
Hint:
We need to find the speed at which we get greatest gas mileage , which is the value of for which the function M(s) is maximum.
The second derivative test to find extremum of F (x) is as follows:
C O and a point x = c is called local maxima if 
0, then the test fails.
The correct answer is: 48
Given,
Gas mileage is denoted by M(s), in miles per gallon,
Speed of the car is denoted by s miles per hour.
The relation between gas mileage and speed is given by
We need to find the value of for which M(s) is maximum.
We use the second derivative test.
First, we find 
That is
TO find critical points,
Which gives
Solving for , we have
The only critical point, we get is s = 48 . So, we check if M (s) has a maximum at s = 48
Now
So 
Thus, M (s) has a maximum at s = 48
The correct option is B)
Note:
We could have also used the first derivative test to find the local
maxima. We need to find critical points and then check whether the
function is increasing or decreasing on either side of the critical
point. If the function is increasing , i. e 
., on the left side
and the function is decreasing, i. e 
., on the right hand side, then we say that is a local maxima.
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