Question

For a certain​ automobile,

7.8M(x)=−.015x² +1.21x−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.

Answer 1

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

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=
`(-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.