Answer: (a) To find the absolute maximum miles per gallon, we need to find the maximum value of M(x). We can do this by finding the vertex of the parabola represented by M(x).
The x-coordinate of the vertex is given by:
x = -b / (2a)
where a = -0.015 and b = 1.32 in our case. Plugging in these values, we get:
x = -1.32 / (2(-0.015)) = 44
So the maximum miles per gallon occurs at a speed of 44 miles per hour.
To find the value of M(x) at this speed, we plug in x = 44:
M(44) = -0.015(44)^2 + 1.32(44) - 7.4 ≈ 32.36
Therefore, the absolute maximum miles per gallon is approximately 32.36, and it occurs at a speed of 44 miles per hour.
(b) To find the absolute minimum miles per gallon, we need to find the minimum value of M(x). We can do this by noting that the coefficient of the x^2 term is negative, which means that the parabola opens downward and has a maximum, so there is no absolute minimum.
We can also confirm this by finding the x-coordinate of the vertex, which we already calculated in part (a) to be x = 44. This means that the parabola has a minimum value of M(44), which we found to be approximately 32.36. However, this is not an absolute minimum, as there are values of M(x) that are smaller than 32.36 for other values of x. Therefore, there is no absolute minimum miles per gallon.
Explanation: