Problem

A W-shaped quartic has valleys near \((-2,~-3)\) and \((3,~-2)\), with a peak near \((0,~4)\). State the turning-point count and order, classify each turn, describe the direction changes, and check the degree bound.

Prompt quartic w-shaped curve with turn classifications withheld.

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Count/order: 3 turns—maximum (−2,−3), maximum (0,4), maximum (3,−2). Trend: decreasing→increasing→decreasing→increasing. Degree bound: a quartic allows at most 3 turns.

Count/order: 2 turns—minimum (−2,−3), maximum (0,4). Trend: decreasing→increasing→decreasing with no final reversal. Degree bound: a quartic allows at most 3 turns.

Count/order: 4 turns—minimum (−2,−3), maximum (0,4), minimum (3,−2), then maximum. Trend: four reversals. Degree bound: a quartic allows 4 turns.

Count/order: 3 turns—minimum (−2,−3), maximum (0,4), minimum (3,−2). Trend: decreasing→increasing→decreasing→increasing. Degree bound: a quartic allows at most 3 turns.

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A W-shaped quartic has valleys near \((-2,~-3)\) and \((3,~-2)\), with a peak near \((0,~4)\). State the turning-point count and order, classify each turn, describe the direction changes, and check the degree bound.

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