Where Extremes Can Hide, and Where They Do

You'll learnwhen a highest and lowest value are guaranteed to exist, and which points are even worth checking for them.

Two questions hide in this topic. Does a highest value have to exist at all? And if it does, where could it be? One theorem answers the first, one short list answers the second, and neither one is a verdict. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THREE CONDITIONS FOR EXISTENCE · TWO WAYS TO BE CRITICAL · v2

f(x) = 3x⁴ + 4x³ − 12x² on [−3, 2] · critical at −2, 0, 1 · extremes at 2 and −2

Before you start

What you're looking at

One polynomial, f(x) = 3x⁴ + 4x³ − 12x², drawn on the closed interval from −3 to 2. It has three flat tangents and two endpoints.

The question

Does a highest value have to exist? And which points are even worth checking?

Step 1 / 12