Maths-
General
Easy

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  20 less or equal than s less or equal than 75.
M left parenthesis s right parenthesis equals negative 1 over 24 s squared plus 4 s minus 50
According to the model, at what speed, in miles per hour, does the car obtain its greatest gas mileage?

  1. 46
  2. 48
  3. 50
  4. 75

hintHint:

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 f to the power of ′′ left parenthesis c right parenthesis less than 0 text . If  end text f to the power of ′′ left parenthesis c right parenthesis equals0,  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
    M left parenthesis s right parenthesis equals negative 1 over 24 s squared plus 4 s minus 50
    We need to find the value of  for which M(s) is maximum.
    We use the second derivative test.
    First, we find M to the power of straight prime left parenthesis s right parenthesis
    M to the power of straight prime left parenthesis s right parenthesis equals negative 1 over 24 times 2 s plus 4 plus 0
    That is
    M to the power of straight prime left parenthesis s right parenthesis equals negative s over 12 plus 4
    TO find critical points,
    M to the power of straight prime left parenthesis s right parenthesis equals 0
    Which gives
    negative s over 12 plus 4 equals 0
    Solving for , we have
    s over 12 equals 4 not stretchy rightwards double arrow s equals 48
    The only critical point, we get is s = 48 . So, we check if M (s)  has a maximum at s = 48
    Now
    M to the power of ′′ left parenthesis s right parenthesis equals negative 1 over 12 text  for all  end text s
    So M to the power of straight prime left parenthesis s right parenthesis less than 0 text  for  end text s equals 48
    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 M to the power of straight prime left parenthesis S right parenthesis greater than 0.,  on the left side
    and the function is decreasing, i. e M to the power of straight prime left parenthesis s right parenthesis less than 0.,  on the right hand side, then we say that  is a local maxima.

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