Definition: f is strictly increasing on I if x1 < x2 implies f(x1) < f(x2). Strictly decreasing if x1 < x2 implies f(x1) > f(x2).

Test: For differentiable f on interval I:

• f'(x) > 0 for all x in interior of I => f strictly increasing on I
• f'(x) < 0 for all x in interior of I => f strictly decreasing on I
• f'(x) >= 0 (with equality only at isolated points) => f still strictly increasing

Important subtlety: f'(x) = 0 at isolated points does NOT prevent strict monotonicity. Example: f(x) = $x^3$ is strictly increasing on R even though f'(0) = 0.

Method to find intervals:

1. Compute f'(x)
2. Find all x where f'(x) = 0 (roots) or f'(x) DNE
3. These points divide the real line into intervals
4. Test the sign of f' in each interval (pick a test point)
5. f is increasing where f' > 0, decreasing where f' < 0

Parametric problems: "Find a such that f(x) = $x^3$ + $ax^2$ + 3x + 5 is increasing for all x." Solution: f'(x) = 3$x^2$ + 2ax + 3 >= 0 for all x. This quadratic is always non-negative iff discriminant <= 0: 4$a^2$ - 36 <= 0, so |a| <= 3.