Direction Comes From the Slope, Never the Height

You'll learnto tell where a function rises and falls from the sign of its derivative, and which points have to be marked before you test.

A function rises where its derivative is positive. Not where the function is positive, and not where the curve you happen to be looking at is climbing. Almost every mistake in this topic is a sign chart built from the wrong function. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · MARK f′ = 0, MARK WHERE f′ FAILS, MARK THE GAPS, THEN TEST · v2

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. Every shaded band is painted from the sign of the curve in that panel.

The question

Which x values split the number line, and what do you test once they are marked?

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