Direction Is a Fact About the Slope, Never About the Height

You'll learnto build an increasing/decreasing chart from f′ and only f′ — marking every place f′ is zero, every place f′ fails to exist, and every gap in the domain of f, before you test a single interval.

This is the shortest rule in Unit 5 and the most reliably misapplied. A function increases where its derivative is positive. Not where the function is positive, not where the function is large, and not where the curve you happen to be looking at is climbing. Almost every error in this topic is a sign chart built from the wrong function. The rest come from a mark that never reached the chart at all — a point where f′ fails to exist, or a gap in the domain of f that splits an interval without being a critical point.

8 STEPS · 6 QUICK CHECKS · MARK f′ = 0, MARK WHERE f′ FAILS, MARK THE GAPS, THEN TEST · v1

f′ > 0 ⇒ f increasing · f′ < 0 ⇒ f decreasing · the chart is built from f′ alone

Before you start

What you're looking at

A function on top and its derivative underneath, on one shared x axis, so a vertical line cuts both at the same input. Every shaded band you see is painted from the SIGN of the curve in that panel — never typed in beside it.

The question

Which x values split the number line, and what do you test once they are marked? One of those two questions has three answers, and dropping any of them deletes a whole interval.

Watch for

The two shadings almost never break in the same places. The top panel's bands answer "where is the graph"; the bottom panel's bands answer "which way is it going", and only the second one is the chart this topic asks for.

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