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

You'll learnto classify a critical point by reading the sign of f′ just before and just after it — to get the direction the right way round, and to recognise that no change at all is a finished answer rather than a stuck one.

Topic 5.2 handed you a list of candidates and Topic 5.3 handed you a sign chart. This is where the two meet, and the whole test fits in one sentence: at a critical point, look at the sign of f′ just before and just after. Negative then positive is a valley. Positive then negative is a peak. And the sign can simply fail to change, which is a finished answer rather than a stalled one — a flat tangent is a rate, not a turn. Notice what the test never asks for: the value of f′ at the point itself. That is why it settles corners and cusps, and why it never comes back inconclusive.

8 STEPS · 6 QUICK CHECKS · READ THE SIGN OF f′ LEFT TO RIGHT ACROSS c · v1

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 f′(x) = 12x(x + 1)(x − 2). Both panels share the same x axis, so a vertical line cuts them at the same input.

The question

At each place where f′ is zero, what does the sign of f′ do as you walk across from left to right — and what does that make the point for f?

Watch for

The direction. Positive then negative is rising then falling, which is a peak. Getting those two words in the wrong order is the single most common error in this topic.

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