Concept preview

What you will be able to do

1. 01Explain and apply the monotonicity test: $f$ is increasing on an interval where $f'>0$ and decreasing where $f'<0$, so the SIGN OF THE DERIVATIVE — not the sign of $f$ — decides the direction.
2. 02Build a sign chart of $f'$ by finding the critical numbers (where $f'=0$ or $f'$ is undefined), testing one point in each interval they create, and recording whether $f$ rises or falls.
3. 03State the intervals of increase and decrease in correct interval notation, and read that same information directly off a graph of $f'$ (positive above the axis, negative below).

Here is the governing theorem, stated plainly. If $f$ is differentiable on an open interval and $f'(x)>0$ for every $x$ in that interval, then $f$ is increasing there; if $f'(x)<0$ throughout, then $f$ is decreasing there. The reasoning is intuitive: $f'(x)$ is the slope of the tangent line, and a positive slope points up-and-to-the-right while a negative slope points down-and-to-the-right. As you sweep across the interval, the little tangent arrows all tip the same way, and the curve as a whole must follow them. (The rigorous justification comes from the Mean Value Theorem you met earlier in Unit 5, but the tangent-arrow picture is what you should carry in your head.)