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)

step-by-step explanation: