The fuel efficiency for a certain midsize car is given by E(v)= -0.016v^2+1.408v+3.8, where E(v)is the fuel efficiency in miles per gallon for a car traveling v miles per hour.

What speed will yield the maximum fuel efficiency? _______mph.
What is the maximum fuel efficiency for this car? _____mpg

To find the speed that will yield the maximum fuel efficiency, we need to find the maximum point of the given quadratic function E(v) = -0.016v^2 + 1.408v + 3.8.

The maximum point of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic function in the form of ax^2 + bx + c.

In this case, the coefficient of v^2 is -0.016, and the coefficient of v is 1.408. So we have a = -0.016 and b = 1.408.

Calculating the x-coordinate of the vertex:
x = -b / (2a)
x = -1.408 / (2 * -0.016)
x ≈ -1.408 / -0.032
x ≈ 44

Therefore, the speed that will yield the maximum fuel efficiency is approximately 44 mph.

To find the maximum fuel efficiency, we substitute the value of v (44) into the function E(v):
E(44) = -0.016(44)^2 + 1.408(44) + 3.8

Calculating the result:
E(44) = -0.016(1936) + 62.272 + 3.8
E(44) = -31.008 + 62.272 + 3.8
E(44) ≈ 35.064

Therefore, the maximum fuel efficiency for this car is approximately 35.064 mpg.

To find the speed that will yield the maximum fuel efficiency, we need to first find the derivative of the fuel efficiency function and locate the critical point(s).

1. Find the derivative of E(v) with respect to v:
E'(v) = -0.032v + 1.408

2. Set E'(v) equal to zero and solve for v to find the critical point:
-0.032v + 1.408 = 0
-0.032v = -1.408
v = -1.408 / -0.032
v ≈ 44

Therefore, the speed that will yield the maximum fuel efficiency is approximately 44 mph.

To find the maximum fuel efficiency, substitute this speed into the original function:

E(44) = -0.016(44)^2 + 1.408(44) + 3.8
E(44) = -0.016(1936) + 62.432 + 3.8
E(44) = -30.976 + 66.232 + 3.8
E(44) ≈ 39.056

Therefore, the maximum fuel efficiency for this car is approximately 39.056 mpg.