For a certain automobile, M(x)=−.015x^2+1.42x−7.8, 30≤x≤60, represents the miles per gallon obtained at a speed of x miles per hour. (a) Find the absolute maximum miles per gallon and the speed at which it occurs. (b) Find the absolute minimum miles per gallon and the speed at which it occurs.

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Step-by-step explanation:

A graph can be helpful. It can show the locations of the maximum and minimum.

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The maximum is found at the vertex of the parabola, located where ...

  x = -b/(2a)

  x = -1.42/(2(-0.015)) = 47 1/3

The value M(47 1/3) is ...

  M(47 1/3) = (-0.015(47 1/3) +1.42)(47 1/3) -7.8 = 0.71(47 1/3) -7.8 = 25 121/150

The maximum gas mileage is 25 121/150 ≈ 25.8067 mpg at 47 1/3 mph.

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The minimum will be at one end or the other of the interval. Here, the maximum is farther from the lower end of the domain, so we expect to find the minimum there:

  M(30) = (-0.015(30) +1.42)(30) -7.8 = 0.97(30) -7.8 = 21.3

The absolute minimum gas mileage is 21.3 mpg at 30 mph.