(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 .