(a)

I/D test :

If  is increasing on the interval, then .

If  is decreasing on the interval, then .

At extrima values, the function .

Observe the graph of  :

Local maximum occurs at .

Local minimum occurs at .

Therefore, the function  for .

The function  increases over the intervals  and .

Therefore,  for  and .

The function  decreases in the interval .

Therefore,  for .

(b)

Observe the graph of  :

The function  is a third degree polynomial function with positive leading coefficient.

The derivative of a third degree function is a second degree function means  is a quadratic function.

The derivative of a second degree function is a first degree function means  is a linear function.

Therefore,  for .

 for .

 for .

(c)

Since the function  is a quadratic function, it is increasing on for .

Therefore, the function  increases on . 

(d)

Since the function  is a quadratic function, the function  is negative minimum for .

The rate of change of  at  is less than the rate of change of  for all other values of .

The function  is decreasing at the greatest rate at .

(a)

 for .

 for  and .

 for .

(b)

 for .

 for .

 for .

(c)

The function  increases on .

(d)

The function  is minimum for .

The function  is decreasing at the greatest rate at .