(a)

I/D test :

If  on the interval, then  is increasing on the interval.

If  on the interval, then  is decreasing on the interval.

Observe the graph.

 over the intervals ,  and .

Therefore the function  is increasing over the intervals ,  and .

 over the intervals  and .

Therefore the function  is decreasing over the intervals  and .

(b)

First derivative test :

(i) If  changes from positive to negative at , then  has a local maximum at .

(ii) If  changes from negative to positive at , then  has a local minimum at .

Observe the graph.

 changes from positive to negative at  and .

Therfore the function  has a local maximum at  and .

 changes from negative to positive at  and .

Therfore the function  has a local minimum at  and .

(c)

Concavity test :

If  for all  in the interval, then the graph of  is concave upward on the interval.

If  for all  in the interval, then the graph of  is concave downward on the interval.

Observe the graph.

 is increases over the interval , then  on the intervals.

Therefore the function  has a concave upward over the intervals .

 is decreases over the interval , then  on the intervals.

Therefore the function  has a concave downward over the intervals .

(d)

Inflection points :

Inflection points are the points at which the concavity changes from up to down or down to up.

Observe the graph.

At  the function  changes from decreasing to increasing, then  changes from negative to positive.

Therefore the inflection point is .

(e)

The function is .

Graph :

Such that above all the conditions are satisfied :

(a)

The function  is increasing on ,  and .

The function  is decreasing on  and .

(b)

The function  has a local maximum at  and .

The function  has a local minimum at  and .

(c)

The function  has a concave upward on .

The function  has a concave downward on .

(d) The inflection point is .

(e) Graph of the function is

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