(a)

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 .

(b)

Second derivative test :

Consider is continuous near to .

(i) If and , then has a local minimum at .

(ii) If and , then has a local maximum at .

When and then test is inconclusive, has either local maximum or local minimum at .

If it fails use the first derivative test.

(a) First derivative test determines the absolute and local maximum and minimum points.

(b) Second derivative test determines the local maximum and minimum points.