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.

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.

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

therefore x=

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.

answer: the four apples you took away.

step-by-step explanation: