(a)

Increasing test :

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

Observe the graph.

on the interval and then is increasing on the interval and .

(b)

First derivative test : 

Consider is a critical number of a continuous function . 

(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 .

(iii) If does not change sign at , then has no local maximum or minimum at .

Observe the graph.

has a local maximum at , because changes its sign from positive to negative.

has a local minimum at and because changes its sign from negative to positive.

(c)

Concave upward :

The function is concave up when is increasing in the interval.

The function is concave upward over the interval , and .

Concave downward :

The function is concave down when is decreasing in the interval.

Observe the graph.

The function is concave downward over the interval , and .

(d)

Point of inflection :

If the graph is of the first derivative then the local minimum or maximum will be the inflection points, since the inflection points occurs where the second derivative will be zero.

The derivative of the function has local maximum at .

The derivative of the function has local minimum at .

The point of inflection occur at and .

(a) is increasing on the interval and .

(b) has a local maximum at and local minimum at and .

(c) is concave upward over the interval , and .

    is concave downward over the interval , and .

(d) has inflection points at and .