Question

(i) the interval(s) over which f rises and where it falls;

(ii) the local extrema.

Answer 1

(i) The derivative of a function may be used to determine whether the function is increasing or decreasing on any intervals in its domain.

Case1: If $f'(x) > 0$ at each point in an interval I, then the function is said to be increasing on I. 

Case2: If $f'(x) < 0$ at each point in an interval I, then the function is said to be decreasing on I.

(ii)  A local extrema (or relative extremum) of a function is the point at which a maximum or minimum value of the function in some open interval containing the point is obtained.

(1)First derivative test:

Let f be continuous on an open interval $(a , b)$ that contains a critical x-value.

1) If $f'(x) > 0$ for all x on (a , c) and $f'(x)<0$ for all x on (c , b), then f(c) is a local maximum value.

2) If $f'(x) < 0$ for all x on (a , c) and $f'(x)>0$ for all x on (c , b), then f(c) is a local maximum value.

3) If $f'(x)$ has the same sign on both sides of c, then $f(c)$ not a maximum nor a minimum value.

(2)Second derivative test:

Let $f'$ and $f''$ exist at every point on the interval (a , b) containing c and $f'(c) = 0.$

1) If $f''(c) < 0$ , then $f(c)$ is a local maximum.

2) If $f''(c) > 0$ , the $f(c)$ is a local minimum.