How do we find the interval(s) where f is increasing and decreasing?

If and first we factor (assuming we can factor or we use " factor(2*x^3-9*x^2+10*x-3);" in Maple to do this) into And notice that we have practiced the followings:

1. If for x in (a,b), then f is increasing in (a,b), and
2. If for x in (c,d), then f is decreasing in (c,d).

But when (which amounts in solving an inequality). Great, it is in standard form, we see the answer is which means that f is increasing in

Similarly, if and the answer is when x is in which means that f is decreaing in

 
• Example 1: Suppose f'(x) = (x-1)(x-3)(x-5) (note that this is the derivative function), then f'(x)>0 in (1,3) u (5, infinity), which means the original function f is "increasing" in (1,3) u (5, infinity). Morevoer, since f'(x)<0 in (-infinity, 1) u (3,5), we see that f is decreasing in (-infinity, 1) u (3,5).
• Example 2: Suppose f'(x) = - (x-1)(x-3)(x-7) , then f'(x)>0 in (-infinity, 1) u (3,7) and hence f is increasing in those intervals. On the other hand, f'(x)<0 in (1,3) union (7, infinity), so f is decreasing there.
• Important Questions that you need to answer later in this class.
• Horizontal Tangent