Where Extrema Can Hide, and Where They Do

You'll learnto check all three conditions of the Extreme Value Theorem before claiming an extreme value exists, to tell a relative extremum from an absolute one and a value from a location, and to run both clauses of the critical point definition instead of only solving f′ = 0.

Two questions get run together here and they are not the same. Does an extreme value exist? That is the Extreme Value Theorem, and it needs three conditions. Where could one be? That is the critical point list, and it has two entries per function, not one. Finding a critical point is the start of the argument, never the end of it.

8 STEPS · 6 QUICK CHECKS · THREE CONDITIONS FOR EXISTENCE · TWO CLAUSES FOR THE SEARCH · v1

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 places where the tangent is flat, two endpoints, and exactly one highest and one lowest value on that interval.

The question

Two of them, and they are different. Does an extreme value have to exist at all? And if it does, which places are even worth checking?

Watch for

The highest value on the whole interval sits at the right endpoint, where the graph is still climbing when the interval runs out. Nothing turns around there — and later, two graphs where the tangent behaves dramatically and still nothing turns around.

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