Up Then Down Is a Peak, and the Order Is the Answer

You'll learnto tell a peak from a valley by reading the sign of f′ just before and just after a critical point, in that order.

A flat tangent is a place to look, not an answer. The First Derivative Test reads the sign of f′ just before and just after the point: negative then positive is a valley, positive then negative is a peak, and no change at all is a finished answer too. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · READ THE SIGN OF f′ LEFT TO RIGHT ACROSS c · v2

f′ > 0 then f′ < 0 ⇒ MAXIMUM · f′ < 0 then f′ > 0 ⇒ MINIMUM · no change ⇒ NEITHER

Before you start

What you're looking at

One function, f(x) = 3x⁴ − 4x³ − 12x² + 5, drawn above its own derivative on a shared x axis, so a vertical line cuts both at the same input.

The question

At each place where f′ is zero, what does its sign do as you walk across from left to right?

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