F(x) = x3 − 9x2 − 21x + 8
(a) Find the interval on which f is increasing.
Find the interval on which f is decreasing
(b) Find the local minimum and maximum values of f.
(c) Find the inflection point.
(d)Find the interval on which f is concave up.
(e)Find the interval on which f is concave down.

Part a)

The given polynomial function is

The first derivative is :

This is the slope function.

On an increasing interval, the slope is greater than 0.

This gives;

Similarly the function is decreasing when

This gives:

Part b)

At fixed points,

This implies that:

We apply the second derivative test.

Hence (-1,19) is a local maximum.

This means (7,-237) is a local minimum.

Part c)

At the point of inflection, the second derivative is zero.

This implies that;

We solve for x to get:

The point of inflection is (3,-109)

Part d)

The given polynomial function is concave up on the interval where f''(x)>0.

This means the interval of concavity is on 6x-18>0.

6x>18

x>3

The function is concave up on (3,∞).

Part e)

The given function is concave down when f''(x)<0.

This implies that;

6x-18<0.

6x<18

x<3

The interval on which the function is concave down is (-∞,3)

a

step-by-step explanation:

when a number is between two straight lines it means the absolute value.

to find absolute value you need to know how many numbers is that number away from 0.

when finding absolute value the number will always come out positive

so absolute value of -3 is 3

absolute value of 3/2 is 3/2

absolute value of -2 is 2

absolute value of 3.5 is 3.5

absolute value of -1 is 1

now put these numbers in order:

a)   |-1|, |3/2|, |-2|, |-3|, |3.5|

the 8 in 8,420 is in the thousands place so it's value is 8 thousand.

the 8 in 6,870 is in the hundreds place so it's value is 8 hundred.

divide 8000 by 800 to find the answer:

8000 / 800 = 10

the 8 in 8,420 is 10 times greater than the 8 in 6,870