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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:
- If for x in (a,b), then f is increasing in (a,b), and
- 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
Wei-Chi Yang
Sat Oct 19 00:19:09 EDT 1996