Question

For a certain​ automobile,

Answer 1

Answer:

The answer is

a) 2,128 miles per gallon at 40,36 miles per hour

b) 1,388 miles per gallon at 60 miles per hour

Step-by-step explanation:

Let's find the extreme points of M(x) for 30≤x≤60. M(x) simplified by dividing to 7.8.

$M(x) = -1,92.10^-3x^2+0,155x-1$

To find extreme points we find tangent of M(x) and equal it to 0.

$M'(x) = -3,84.10^-3x+0,155=0$

therefore x=$\frac{-0,155}{-3,84.10^-^3} = 40,36$

This divides our x to 2 different range 30≤x≤40,36 and 40,36≤x≤60

Now we pick a number and investigate from the ranges

30≤x≤40,36   →    x = 35    →    M'(35) = 0,0206 which is > 0 and that means M is increasing

40,36≤x≤60   →    x = 45   →    M'(45) = -0,0178 which is < 0 and that means M is decreasing

Next let's look at critical points

   x                  M(x)          Before              After             Decision

  30                1,922              -                       ↑             Local Minimum Point

  40,36           2,128             ↑                       ↓             Local Maximum Point

  60                1,388             ↓                       -              Local Minimum Point

As we can see (30, 1,922) and (60, 1,388) are local minimum points and (40,36 , 2,128) is the local maximum point.

Since (60 , 1,388) is the lowest local minimum point it is the absolute minimum point and (40,36 , 2,128) is the absolute maximum point.