Consider the function f ( x ) = − 2 x 3 + 27 x 2 − 108 x + 9 . for this function there are three important open intervals: ( − [infinity] , a ) , ( a , b ) , and ( b , [infinity] ) where a and b are the critical numbers. find a and b

A=3 and B=6

Step-by-step explanation:

Increasing and Decreasing Intervals of Functions

Given f(x) as a real function and f'(x) its first derivative.

If f'(a)>0 the function is increasing in x=a

If f'(a)<0 the function is decreasing in x=a

If f'(a)=0 the function has a critical point in x=a

As we can see, the critical points may define open intervals where the function has different behaviors.

We have

Computing the first derivative:

We find the critical points equating f'(x) to zero

Simplifying by -6

We get the critical points

They define the following intervals

Thus A=3 and B=6

256π units²   (answer b)

step-by-step explanation:

the formula for the surface area of a sphere of radius r

is             a = 4πr²

and so, if r = 8 units, then the surface area is:

a = 4π(8 units)² = 256π units²   (answer b)