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)

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.